David Cameron has said he is a "very proud dad" after his wife Samantha gave birth to a girl, their fourth child, according to BBC.
The baby - weighing 6lb 1oz - was delivered by caesarean section while the family was on holiday in Cornwall.
Mr Cameron said the new arrival was "absolutely thrilling" and added that the couple had yet to decide on a name.
The baby had been expected in September. The Camerons have two children and their eldest son Ivan died last year.
Mrs Cameron gave birth at the Royal Cornwall Hospital in Truro at midday after going into labour on Tuesday morning.
Speaking to journalists outside the hospital, Mr Cameron said the couple had decided to go to the hospital after his wife had begun having contractions.
"It was very exciting, we were on holiday, thought we were going to get through the holiday and then have a baby, and then it just seemed to kick off a bit quickly," he said.
"It all happened very very quickly. It is absolutely thrilling, really exciting and it`s just lovely they are both doing so well."
Describing his daughter as "unbelievably beautiful", he thanked staff at the hospital and joked that his role during the past few hours had been to "make the toast and the tea".
He said Samantha would choose the baby`s name, suggesting she was already looking through a book of names, and added that it may well have a local connection.
"It was very exciting having a baby in Cornwall. We`ll have to have something Cornish in there somewhere," he said.
Bookmakers William Hill have opened betting on the choice of name, saying Marnie, Enid, Annabel and Elizabeth are among the most popular suggestions with the public so far.
In a statement, No 10 said the couple "would like to thank the doctors and nurses at the hospital for their help and kindness".
Mr Cameron has said he will take paternity leave but it is not clear whether this will begin immediately.
While the Camerons have been on holiday, deputy prime minister and Lib Dem leader Nick Clegg has been in charge of government business.
Offering his congratulations to the prime minister, Mr Clegg said: "Miriam and I are absolutely delighted by the fantastic news.
"All our best wishes are with David and Samantha at this special time for them and their family."
Labour leadership contender David Miliband also offered the couple his congratulations.
It is the second time since 2000 that the wife of a sitting prime minister has given birth.
When Cherie Blair gave birth in May 2000 - while her husband was in office - it was the first time a serving PM`s spouse had given birth for more than 150 years.
The Camerons already have two young children, Nancy and Arthur Elwen, six and four years old respectively.
Their first child, Ivan, who was born profoundly disabled and needed round the clock care, died in February 2009.