David Cameron's wife Samantha remains in hospital following the birth of the couple's new baby.
She and the Prime Minister named their daughter Florence Rose Endellion.
Endellion refers to the Cornish village of St Endellion, close to where they are holidaying and which Mr Cameron has visited over the years.
The couple received congratulatory messages from the Queen, the Prince of Wales, Gordon and Sarah Brown and French President Nicolas Sarkozy.
Downing Street said the couple's two elder children, six-year-old Nancy and four-year-old Arthur Elwen, visited their sister for the first time.
Flowers were delivered to the hospital to welcome the latest addition to the Cameron family.
According to the most recent statistics, Florence is the 94th most popular girl's name and was given to 608 babies born in 2008.
But it recently soared in popularity after pop sensation Florence and the Machine stormed the charts and charmed the nation. Mr Cameron proclaimed himself a fan of the flame-haired singer.
The Camerons' choice of middle name was met with delight in the village of St Endellion.
Prebendary John May, who lives at the rectory, said it was "brilliant news" that the Prime Minister and his wife were inspired by the village.