David Cameron's new-born daughter is to be named Florence Rose Endellion, Downing Street has revealed.
The baby was born by caesarean section, weighing 6lb 1oz, on Tuesday while the prime minister and wife Samantha were on holiday in Cornwall.
St Endellion is a village in the north of the county.
Mrs Cameron and baby Florence are expected to remain in hospital until the weekend. The Camerons already have two children, Nancy and Arthur.
The Camerons' first child, Ivan, who was born profoundly disabled and needed round the clock care, died in February last year.
The new baby had been expected in September, but was born at the Royal Cornwall Hospital in Truro after Mrs Cameron went into labour on Tuesday morning. Mr Cameron spent Tuesday night at the hospital, it has been confirmed.
After the birth, the prime minister said: "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.
"It all happened very very quickly. It is absolutely thrilling, really exciting and it's just lovely they are both doing so well."
He described baby Florence as "unbelievably beautiful".
Downing Street said the couple had received congratulatory messages from the Queen, the Prince of Wales, former Prime Minister Gordon Brown and wife Sarah and French President Nicolas Sarkozy.
US President Barack Obama also called Mr Cameron to offer his congratulations before the men briefly discussed the Middle East, a Downing Street spokesman said.
Nancy, six, and Arthur, four, have now visited their sister.
Bookmaker William Hill said that no customer had placed a bet on Florence being the baby's name, which was offered at odds of 16/1.
No one had plumped for Endellion either, the firm added.
Elizabeth Uglow, clerk of St Endellion Parish Council, told the BBC: "It's an honour to have their baby named after us."
She said: "It's delightful news and I send my best wishes to David and Samantha Cameron."
It is the second time since 1850 that the wife of a sitting prime minister has given birth, following Cherie Blair in May 2000.