Jeremy Hunt and Boris Johnson will face 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 after several rounds of voting.
Johnson is the clear frontrunner, consistently topping the politicians’ ballots by a wide margin.
Theresa May, who resigned as party leader on June 7, will remain as prime minister until the outcome of the membership vote, which is expected later in July.
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.
Both contenders say they want to re-negotiate 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.
The terms of the latest extension also explicitly excluded this — although Brussels has indicated that the non-binding Political Declaration on future ties can be modified.
Here is what the two remaining candidates for the UK's top job say about Brexit:
1. Boris Johnson
EU referendum vote: LEAVE. Round 1: 114 votes. R2: 126. R3: 143. R4: 157. R5: 160.
The former foreign secretary and mayor of London is the clear frontrunner for the leadership contest.
Johnson was a prominent lead figure in the Vote Leave campaign in the months leading up to the 2016 EU referendum, and has maintained that he wants to see the UK leave the European Union regardless of having a deal in place.
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. "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.
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. He has also said he would refuse to pay the UK's estimated €45 billion divorce bill unless the EU offers better terms — a move 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. 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".
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.
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 opposed a hardline approach to negotiations, arguing that it 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.
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.
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 German Chancellor Angela Merkel and French President Emmanuel Macron he noticed there was "willingness to engage".
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 will now be staged among the party membership, who will choose between the final pair. The new party leader, and the UK's next prime minister, is expected to be in place by late July.
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.