She said her goal is to "negotiate the best deal for Britain in leaving the European Union".
"We will have a new prime minister in that building behind me by Wednesday evening", Cameron said in a brief statement outside the leader's official London residence.
David Cameron will chair his final cabinet meeting today and will answer prime minister's questions in the House of Commons for the last time tomorrow.
May was declared the victor of the governing Conservative Party's leadership contest after her only rival, junior Energy Minister Andrea Leadsom, announced earlier on Monday that she was withdrawing from the race. She said, "I am honored and humbled to have been chosen by the Conservative Party to become its leader".
And she paid tribute to Mrs. Leadsom for her "dignity" in withdrawing her leadership bid, as well as to the three other candidates who ran in the contest.
MPs cheered and banged on their desks as the home secretary arrived at what has been one of the most extraordinary and fastest moving days for generations in British politics. "We need to unite our country. we are going to give people more control over their lives and that is how we will build a better Britain", she, who is now the home secretary said in her first speech. Her allies accused May supporters of trying to undermine Leadsom.
Two of the leading Brexiteers, former London Mayor Boris Johnson and Justice Secretary Michael Gove, were briefly Conservative leadership contenders, and May could show magnanimity by putting them in her Cabinet.
But events have been brought forward after Ms May's only opponent, Andrea Leadsom, withdrew from the race to succeed him. May and Cameron were against Brexit, while Leadsom campaigned in its favor.
Leader Jeremy Corbyn has lost the support of most Labour members of Parliament and is facing a challenge from legislator Angela Eagle.
May, 59, launched her campaign for leadership of the Conservative Party in Birmingham, but Leadsom announced her withdrawal from the race soon after in London.
Prior to Leadsom's withdrawal, May gave a speech in which she promised not to undo Britain's referendum decision to leave the European Union and to make Britain a country that will work for everyone.
As energy minister under Cameron, Rudd was seen as efficient and reliable and she backed May early in the leadership contest.
Senior Conservatives were to hold talks later on Monday to give formal approval to May as leader of the party and therefore also the prime minister - a process that under British laws requires no general election.