Cameron’s office announced the birth, which was a few weeks earlier than expected, yesterday afternoon.
A statement from Downing Street said: “Both the baby -- who was born weighing 6lbs 1oz (2.7 kg) -- and Mrs Cameron are doing very well.”
A spokeswoman would not confirm whether Cameron, who is currently on holiday, will take off a full two weeks of paternity leave. The couple have yet to name the baby girl, although Daisy, Lucy and Samantha are the favourites with bookmakers.
Former Prime Minister Tony Blair was the last leader to have a baby while in office after his fourth child, Leo, was born in May 2000.
The Camerons already have two children, Nancy and Arthur.