Mrs May had been expecting a nine-week Conservative leadership race, but rival Andrea Leadsom withdrew on Monday.
Mr Cameron will tender his resignation to the Queen on Wednesday, leaving Mrs May, home secretary since 2010, to appoint her own ministerial team.
Mrs May said she was “honoured and humbled” to be taking over and pledged to make a success of the UK’s EU exit.
Mrs Leadsom’s surprise announcement left Mrs May, who had been the front runner, as the only candidate to take over the leadership of the Conservative Party and, therefore, also become prime minister.
Flanked by dozens of Conservative MPs, Mrs May praised Mr Cameron for his stewardship of the party and the country and paid tribute to Mrs Leadsom for her “dignity” in withdrawing her leadership bid.
But senior Labour MP Jon Trickett has joined the Lib Dems and Green Party in calling for a snap general election.
Mr Trickett, Labour’s general election co-ordinator and an ally of leader Jeremy Corbyn, said it was “crucial” to have a “democratically elected prime minister” and he was putting the party on “general election footing”. Mrs May has rejected such demands.
BBC political editor Laura Kuenssberg said EU negotiation, controlling immigration and managing the economy were “huge issues” that would challenge Mrs May’s desire for a “steady as she goes” approach. (Courtesy BBC)