Theresa May has said she is “honoured and humbled” to have been chosen as the new leader of the Conservative Party and UK prime minister.
Speaking outside Parliament, the PM-in-waiting pledged to build a “better Britain” and make Brexit a “success”.
David Cameron is to tender his resignation to the Queen on Wednesday.
It comes after Mrs May’s only rival to succeed Mr Cameron as Conservative leader, Andrea Leadsom, pulled out of the contest earlier on Monday.
Mrs May paid tribute to Mrs Leadsom for her “dignity” and praised Mr Cameron’s “leadership” of the party and country.
Mr Cameron, PM since 2010, decided to quit after the UK’s Brexit vote.
In a brief statement outside No 10, Mr Cameron said Mrs Leadsom had made “absolutely the right decision” to stand aside and that he was “delighted” Mrs May, the home secretary, was to succeed him.
He said a “prolonged period of transition” was not necessary, and added: “So tomorrow I will chair my last cabinet meeting. On Wednesday I will attend the House of Commons for Prime Minister’s Questions.
“After that I expect to go the Palace and offer my resignation.”
The prime minister praised Mrs May as “strong” and “competent” and he said she was “more than able to provide the leadership” the UK needs in the coming years.
“She will have my full support,” he added.
It comes after another day of dramatic developments in the political world, when Mrs Leadsom pulled out of the two-way leadership contest, leaving Mrs May – the front runner – as the only candidate to take over leading the party and become PM.
She said Mrs May had begun the day launching her leadership campaign to take to the party membership – and within the space of several hours found out she would be prime minister by Wednesday.
Mrs May would now have to decide the make-up of her new cabinet, she said.
Mr Cameron announced his intention to resign as prime minister on 24 June, after finding himself on the losing side of the EU referendum, with the UK voting by 52% to 48% in favour of leaving.