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.
(BBC News)