Cameron said he would offer his resignation to the Queen on Wednesday afternoon at Buckingham Palace.
Mrs May’s only rival to succeed Mr Cameron as Conservative leader, Andrea Leadsom, pulled out earlier on Monday.
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 year.
“She will have my full support,” he added.
BBC chief political correspondent Vicki Young said events were “moving very rapidly”.
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 on June 24 after losing to the Brexit proponents.