May was left as the only contender standing after the withdrawal from the leadership race of Andrea Leadsom, who faced criticism for suggesting she was more qualified to be premier because she had children.
Cameron later announced she would take over from him on Wednesday, when he is expected to go and see Queen Elizabeth II to tender his resignation and recommend Home Secretary May as his successor.
"On Wednesday, I will attend the House of Commons for Prime Minister's Questions and then after that I expect to go to the palace and offer my resignation so we'll have a new prime minister in that building behind me by Wednesday evening," he said in a statement delivered outside 10 Downing Street.
Britain has faced the worst political turmoil in generations following June 23's shock vote to leave the European Union, which prompted Cameron to step down.
His Conservatives have endured a bitter leadership race while the leader of the main opposition Labour party, Jeremy Corbyn, is also facing a challenge to his job.
While May supported Britain staying in the EU, she cut a low profile during the referendum and has insisted she will honour the vote, stressing: "Brexit means Brexit".
"There will be no attempts to remain inside the EU, there will be no attempts to rejoin it by the back door, no second referendum," she said at a campaign event before Leadsom pulled out.
May wants to begin formal talks to leave the EU by the end of the year at the earliest despite pressure from Brussels to speed up.
Jeroen Dijsselbloem, the Dutch finance minister who heads the Eurogroup of his 19 eurozone counterparts, restated calls for the transfer of power to take place as soon as possible.
"The sooner we can sort out -- let me say it diplomatically -- this problematic situation, the better," Dijsselbloem told reporters.
General election call
The pound, which hit a 31-year low in the wake of the Brexit vote, briefly rose after Leadsom, a pro-Brexit figure with no senior ministerial experience, withdrew from contention to be prime minister.
As senior MP Angela Eagle formally launched her leadership challenge against Corbyn, Labour suggested a general election would need to be held soon after May takes office.
"It is crucial, given the instability caused by the Brexit vote, that the country has a democratically elected prime minister," said election coordinator Jon Trickett.
"I am now putting the whole of the party on a general election footing."
May, a 59-year-old clergyman's daughter will be Britain's second female prime minister after Margaret Thatcher, who was in office from 1979-1990.
She has portrayed herself as the leader who can unite the country following a bitterly divisive campaign, and a tough negotiator who can stand up to Brussels in what promise to be tortuous talks over Britain's exit from the European Union.
Leadsom's withdrawal means all the top Brexit campaigners -- Boris Johnson, Michael Gove, Leadsom and outgoing UK Independence Party leader Nigel Farage -- have now stepped back from leadership roles.
"Brexiteers threw rocks through the window, now they're all running away from the house," author Salman Rushdie said on Twitter.
'Howl of pain'
On the latest in a string of recent astonishing days at Westminster, Eagle also launched her bid to take over the leadership of the main opposition Labour Party from veteran socialist Corbyn.
Corbyn has widespread support among party members but has lost the confidence of at least three-quarters of his MPs, many of who accuse him of lacklustre campaigning to stay in the EU.
Eagle, who is from Labour's "soft" left and was the first female MP to enter a civil partnership with her female partner in 2008, said Britain faced "dangerous times".
Of the referendum, she added: "This vote was a message for millions in our country who felt that no-one had listened to them for a very long time".
"For many of them, it was a howl of pain."
By contrast with the Conservative outcome, there is still major uncertainty about how the Labour leadership contest will work.
Central to the race will be a decision by Labour's governing National Executive Committee (NEC) about whether Corbyn automatically gets on to the ballot or needs to secure 51 lawmakers' nominations, which he could struggle to do.
The NEC is due to meet on Tuesday.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)
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