Either Boris Johnson or Jeremy Hunt will become the new Conservative leader shortly when the result of the contest to succeed Theresa May is announced.
The outcome of the ballot of about 160,000 Tory members will be revealed just before midday.
The victor will officially become UK prime minister on Wednesday.
Mr Johnson, former mayor of London, is seen as the clear favourite, although a number of senior figures have said they will not serve under him.
Education Minister Anne Milton tweeted her resignation just half an hour before the leadership result was due to be revealed.
Explaining her decision, she said: “I believe strongly that Parliament should continue to play a central role in approving a deal and that we must leave the EU in a responsible manner.”
Mr Johnson has said the UK must leave the EU with or without a deal on by 31 October.
Mrs May, who is standing down after a revolt by Conservative MPs over her Brexit policy, has chaired her last cabinet meeting.
Theresa May. Photo: AFP
She will officially tender her resignation to the Queen on Wednesday afternoon after taking part in her final Prime Minister’s Questions.
Her successor will take office shortly afterwards, following an audience at Buckingham Palace.
Conservative members have been voting by post for the past two-and-a-half weeks. It is the first time they will have selected a serving prime minister.
Since he made the final two candidates last month, Mr Johnson – who led the Leave campaign in the 2016 Brexit referendum – has been regarded as the frontrunner.
Conservative MP Sir Michael Fallon told the BBC’s Radio 4 Today programme Mr Johnson would “improve” the Brexit deal with the EU in a way that would satisfy Parliament.
“One of the great attractions of Boris taking over our party is that he is optimistic and ambitious,” he added.
Mr Johnson has previously said the withdrawal agreement Mrs May negotiated with the EU is “dead”.
The month-long leadership campaign has been dominated by arguments over Brexit.
Mr Hunt, the foreign secretary, has said he is better placed to secure a negotiated exit and would be prepared to ask for more time beyond the Halloween deadline to finalise it.
Mr Johnson has said he is determined to take the UK out of the EU on 31 October, if necessary without a deal, and all ministers who serve in his cabinet must “reconcile” themselves to this.
The BBC’s political correspondent Nick Eardley said the incoming prime minister would inherit a “daunting in-tray”.
The EU has repeatedly insisted the withdrawal agreement is not up for renegotiation.
Many MPs, including some Conservatives, have also said they will do all they can to stop no deal if the next PM tries to take that route.
In a no-deal scenario, the UK would immediately leave the EU with no agreement about the “divorce” process, and would, overnight, leave the single market and customs union – arrangements designed to facilitate trade.
Chancellor Philip Hammond, Justice Secretary David Gauke and International Development Secretary Rory Stewart have all said they cannot support this and will resign if Mr Johnson is elected.
Mr Gauke said he had held a “very friendly” meeting with Mr Johnson on Monday, in which he told him they had “very different views on the consequences of a no-deal Brexit”.
But speaking on the Today programme on Tuesday, Mr Gauke said he wanted Mr Johnson to succeed – should he become PM – and he would not vote against the Conservative Party in any confidence motion.
Beyond those resignations, there are likely to be wholesale changes in cabinet if Mr Johnson wins. Such a reshuffle will only begin if and when he enters Downing Street on Wednesday.
Sir Alan Duncan, who quit his Foreign Office role on Monday, had called for MPs to have a vote before this on whether they actually back Mr Johnson forming a government.
He said it would show whether Mr Johnson, who like his predecessor will depend on the votes of the Democratic Unionists to form a majority, has “the numbers to govern”.
However, his request was turned down by Commons Speaker John Bercow.