Speaking outside the 10 Downing Street official residence, Mr 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, clearing the way for the female Prime Minister.
In a brief statement, Mr Cameron said 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.