Mr Cameron hailed his successor as “strong and competent” as he announced he would go to Buckingham Palace to offer his resignation to the Queen following Prime Minister’s Questions.
Ms May’s coronation follows rival Andrea Leadsom’s shock decision to pull out of the Tory leadership race.
Ms Leadsom announced she was quitting the contest shortly after apologising to Ms May over an interview in which she appeared to suggest that the fact she was a mother gave her the edge over the childless home secretary as a future prime minister.
Some supporters suggested she had been targeted for attack, after a string of high-profile media stories questioning the accuracy of her CV and demanding that she release her tax returns.
The prime minister said he was “delighted” the home secretary would succeed him in Downing Street.
Mr Cameron will chair his last cabinet meeting this morning and will take his final Prime Minister’s Questions tomorrow at noon.
In a statement in Downing Street yesterday, he said: “We’re not going to have a prolonged Conservative leadership election campaign. I think Andrea Leadsom has made absolutely the right decision to stand aside and it’s clear Theresa May has the overwhelming support of the Conservative parliamentary party.
“I’m also delighted that Theresa May will be the next prime minister. She is strong, she is competent, she is more than able to provide the leadership that our country is going to need in the years ahead and she will have my full support.
“Obviously with these changes we now don’t need to have a prolonged period of transition and so tomorrow I will chair my last cabinet meeting, 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 will have a new prime minister in that building behind me by Wednesday evening.”
The leadership contest had been timetabled to run until September but was dramatically cut short after Ms Leadsom announced she was withdrawing from the race.
Conceding she had too little support among Tory MPs to deliver a “strong and stable government”, the energy minister said she had concluded that “the interests of our country are best served by the immediate appointment of a strong and well-supported prime minister” and was offering Ms May her “full support”.
Ms May was facing demands for an immediate general election after securing her place as the UK’s second female prime minister.
And, crucially, the figureheads of the Brexit campaign in last month’s referendum, Michael Gove and Boris Johnson, offered the home secretary their backing as new Conservative leader.
Labour election co-ordinator Jon Trickett said he was putting the whole party on general election footing, on the very day when its own leadership contest was kicked off by a formal challenge to Jeremy Corbyn from Angela Eagle.
And the Liberal Democrats and Greens demanded an early election following Ms May’s “coronation”.
Mr Brady said both Downing Street and Buckingham Palace had been “properly consulted and involved” in the process.
Meanwhile, Ms Leadsom was the victim of smears and faced an “onslaught” of personal attacks during her failed leadership bid, allies have claimed.
Campaign manager Tim Loughton criticised the use of “spin and underhand tactics” and attacked colleagues who had chosen to “further their own ends”.
MPs claimed Mrs Leadsom had endured a “sledging” at the hands of opponents who wanted to stop her taking the keys to No 10. But Tory former Cabinet minister Eric Pickles insisted the criticism she had faced did not compare to the challenges she would have encountered as prime minister.
Mr Loughton hit out at the media and Tory colleagues in a statement after Ms Leadsom announced she was pulling out of the race.
He said: “Throughout our short campaign we have made it clear that we should be relentlessly focused on the positive case for electing Andrea as leader without the need to undermine the qualities of her opponents.
“Despite an onslaught of often very personal attacks from colleagues and journalists we have never deviated from that goal. Colleagues who have chosen to further their own ends by putting smear above respect will no doubt account for their motivations but it is genuinely puzzling to understand who they think they are helping. It is certainly not our party or our constituents.
“It is absolutely not the job of media commentators to ‘big up’ politicians whether in this leadership contest or elsewhere in politics. But neither should it be their compulsion constantly to try to trip them up. Using spin and underhand tactics against decent people whose prime motivation is to serve has for too long undermined the confidence of the public in our politics. This need not be inevitable. It is this much needed fresh start to how we do our politics that was to be the centre piece of Andrea’s campaign and which we must together progress whatever her role in the future.”