Jeremy Hunt and Boris Johnson are facing a run-off vote of the Conservative membership to determine who becomes party leader and the new UK prime minister.
The current foreign secretary and his predecessor were selected by the party’s Members of Parliament (MPs) on June 20 after several rounds of voting.
Johnson is the clear frontrunner, consistently topping the politicians’ ballots by a wide margin.
The winner will have three months to tackle the burning issue of the day: Brexit. The outcome will determine the government's approach to the United Kingdom's scheduled departure from the European Union on October 31.
The original departure date was extended twice after the negotiated divorce agreement was repeatedly rejected by the British parliament.
Renegotiate — or no deal
Both contenders say they want to renegotiate the deal with the EU but are ready to leave without one if necessary, with Johnson taking a harder stance over sticking to the deadline.
European leaders have repeatedly ruled out re-opening the Withdrawal Agreement on the UK's exit terms, struck by London and Brussels and approved by the other 27 EU countries in November 2018.
The terms of the latest extension also explicitly excluded a re-negotiation — although Brussels has indicated that the non-binding Political Declaration on future ties can be modified.
Under a no-deal Brexit, the UK's trading relationship with the EU and with much of the rest of the world would change. The UK would leave the EU's single market and customs union and begin trading with the bloc under World Trade Organization (WTO) rules — while EU deals with many other countries would no longer apply to the UK. Legal arrangements covering many other aspects of everyday life would also abruptly cease to apply.
Here is what the two remaining candidates for Britain's top job say about how they plan to get the UK out of the EU:
1. Boris Johnson
EU referendum vote: LEAVE. MPs' vote — Round 1: 114 votes. R2: 126. R3: 143. R4: 157. R5: 160.
The former foreign secretary and mayor of London was a prominent lead figure in the Vote Leave campaign in the months leading up to the 2016 EU referendum. He has maintained that he wants to see the UK leave the European Union regardless of whether a deal is in place.
'Do or die'
Speaking on the day May announced her departure (May 24), Johnson said: "We will leave the EU on October 31, deal or no deal. The way to get a good deal is to prepare for a no deal."
He repeated the message when he officially launched his campaign on June 12, adding that if parliament tried to block it, "we will wreak the whirlwind and face mortal retribution from the electorate". Only when Brexit had been delivered, he argued, could the country and society be united.
But he denies he is aiming for a no-deal outcome — even asserting that the chances of it happening are a "million to one". "I don't think that we will end up with any such thing," he said at the launch. "But it is only responsible to prepare vigorously and seriously for no deal."
Later, in a live TV debate among the leadership hopefuls, he said a departure on October 31 was "eminently feasible" but offered no firmer commitment despite questioning. Some observers suggested he might be softening on no-deal — which his supporters denied.
'We can renegotiate'
Boris Johnson has also said the Irish backstop must be renegotiated in any new deal with the EU — something EU leaders have consistently ruled out. The insurance policy designed to guarantee an open border on the island of Ireland incurred the wrath of Brexiteers, ensuring the deal's downfall in parliament. But Ireland and the EU insist it cannot be ditched.
In an interview with the BBC, Johnson said it would be possible to "take the bits of the current withdrawal agreement... that are serviceable and get them done". EU leaders have remained firm in their opposition to "cherry-picking" from the bloc's rules and say the negotiated deal cannot be carved up.
The Northern Irish border question, he said, could be tackled during the "implementation period" after the UK's exit. The negotiated transition period, however, is part of the withdrawal agreement. The EU is adamant this must be passed for it to take effect. Johnson said he believed there would be a new approach as "politics has changed so much since 29 March".
He argues that technology can help maintain an open border, though many trade experts say nowhere in the world have such solutions yet brought a completely open frontier between two different regulatory and customs regimes.
Last year Johnson's comparison of the Irish border to the boundaries between London boroughs was widely ridiculed.
Free trade — even with no deal
The EU has made it clear that under a no-deal Brexit, the UK would be treated as a "third country". The European Commission's contingency plan says this means levying duties and taxes, and carrying out controls.
However, Boris Johnson — and key supporters — argue otherwise. "There will be no tariffs, there will be no quotas, because what we want to do is to get a standstill in our current arrangements under GATT 24," he told a leadership campaign debate.
Article 24 of the World Trade Organisation's General Agreement on Tariffs and Trade (GATT) has previously been invoked by Brexiteers. It allows for smooth trade in goods — but not services — once a free trade deal has been agreed in principle.
But this would require the agreement of both sides. The EU has repeatedly said it would not apply in a no-deal Brexit — which by definition, would imply a failure to agree on key issues.
Johnson's view has been flatly rejected by many trade experts, as well as the Bank of England governor Mark Carney, fellow Brexiteer the UK's international trade secretary Liam Fox, and even by one of his supporters, the attorney-general Geoffrey Cox.
The favourite to become prime minister has also said he would refuse to pay the UK's estimated €45 billion divorce bill unless the EU offers better terms — using the phrase "creative ambiguity" over whether payments would be made. Such a move regarding the UK's legal obligations has been criticised as reckless and dangerous.
The 54-year-old was long thought to be a frontrunner to succeed UK Prime Minister David Cameron in 2016, but he famously pulled out of the race after his Vote Leave ally Michael Gove derailed his efforts.
Foreign secretary for two years in May's government, he resigned in protest at her so-called "Chequers plan" for the UK's future trading relationship with the EU.
2. Jeremy Hunt
EU referendum vote: REMAIN. MPs' vote — Round 1: 43 votes. R2 46. R3: 54. R4: 59. R5: 77.
British Foreign Secretary Jeremy Hunt entered the leadership contest with a firm message posted to his Facebook page, outlining his beliefs of healing a "fractured" Conservative party and avoiding a general election that would be a "prospectus for disaster".
'A different deal'
Hunt voted three times in favour of the Brexit deal negotiated by Theresa May, but said that it was only because he believed that it was better than a no-deal scenario.
But should he win the leadership contest, he says he would pursue a reopening of negotiations with the European Union as he believes a different deal is "the only solution". Launching his leadership bid, he stated that "tough negotiation" and not "empty rhetoric" was needed to deliver Brexit.
The foreign minister does not rule out leaving the EU without a deal if it were the only way to bring Brexit about, but is not committed to October 31 as a hard deadline.
No-deal 'with a heavy heart'
The foreign secretary has issued strong warnings against a no-deal Brexit, while also saying he would be ready to accept it — leading to accusations that he is prone to "flip flop" on the issue, and that he is succumbing to pressure to woo Tory hardliners to eat into Johnson's lead.
Hunt challenged Johnson during the TV debate, asking him what message he would deliver to a farmer facing 40 percent export tariffs in the event of no-deal.
He has also described no-deal as "political suicide" for the Conservative Party. His argument against a hardline approach to negotiations is that this would be met harshly in Europe and could trigger a general election in the UK, which could see the opposition in government "by Christmas". He has promised that he would not trigger any action to provoke a general election before delivering Brexit.
However, more recently Jeremy Hunt has said he would be ready to take the UK out of the EU without a deal on October 31, even if that meant businesses going bankrupt. "I would do so but I would do so with a heavy heart," he told the BBC.
He has published a 10-point Brexit preparation plan he would implement as prime minister: including ramping up no-deal preparations, establishing a no-deal cabinet task force, plans to keep trade flowing, a relief programme to compensate farming and fishing, and funds to develop customs solutions. A final decision on whether there was a "realistic chance" of a successful new deal would be taken on September 30.
A former culture and health minister, the foreign secretary emphasised several times that his negotiating skills made him the right candidate to reach an agreement with the European Union on Brexit.
"With me to face the unyielding Brussels machine, you would be sending a prime minister that has been negotiating all his life," he said.
Hunt added that in his conversations with European leaders such as Angela Merkel and Emmanuel Macron he noticed there was "willingness to engage". However, the French president has reportedly described any attempt to re-negotiate the current deal as a "non-starter", while the German chancellor backed the official EU stance at the European Council summit in June.
While Hunt campaigned for the UK to remain in the EU during the leadup to the 2016 referendum, he has since delivered scathing remarks toward the bloc, including a comparison to a "prison" of the Soviet Union.
Watch Hunt's speech below.
How will the new leader be chosen?
The Conservative Party selects its new leader in two stages: a series of votes among its MPs, followed by a ballot of party members.
MPs carried out five rounds of voting between June 13 — 20. The initial candidate list of ten was reduced until only two remained.
A run-off is now being staged among the party membership, who will choose between the final pair. Theresa May, who resigned as party leader on June 7, remains as prime minister until the outcome of the membership vote. The result is expected on July 23.
In early 2018 the Conservatives had an estimated 124,000 members, a figure which is reported to have risen to more than 160,000 by May 2019.
The increase is partly due to a recruitment drive, but there have also been reports of anti-EU hardliners and some pro-Europeans joining the party in recent months, to try to influence its direction over Brexit.
A recent survey by YouGov of Tory party members found that a large majority would be prepared to accept significant damage to the economy, the break-up of the UK, and the destruction of their party — in order to deliver Brexit.
What happened to the other candidates?
The final round of voting among Tory MPs saw the elimination of Environment Secretary Michael Gove, a Brexiteer most famous for shattering the leadership hopes of his longtime friend and Vote Leave ally Boris Johnson after the EU referendum in 2016.
In the previous round Home Secretary (interior minister) Sajid Javid was knocked out. He had said his focus would be on getting a new deal with the EU, adding that if at the end of October there was a choice between no deal and no Brexit, he would choose no deal.
Earlier, International Development minister Rory Stewart — the only candidate who vehemently opposed a no-deal Brexit — was voted out. He had strongly criticised his leadership rivals over their stance, defending Theresa May's deal as a sensible compromise — and became known for his innovative campaigning methods, using walkabouts and social media
Former Brexit minister and no-deal hardliner Dominic Raab was eliminated in the second ballot. Health minister Matt Hancock got through the first round but withdrew from the race afterwards.
Three candidates were eliminated in the first round of voting among MPs: former minister Esther McVey, former leader of the House of Commons Andrea Leadsom, and Conservative lawmaker Mark Harper.
Three other leadership contenders — James Cleverly, Kit Malthouse and Sam Gyimah — pulled out of the race before the first round, following criticism that there were too many candidates, and the party's selection rules were tightened.
This article has been updated since the original publication on May 28.