David Cameron and his wife Samantha are considering what to call their new daughter, with the Prime Minister keen to include a reference to the county of her birth.
Mr Cameron said he was open to suggestions for Cornish names after the couple's "unbelievably beautiful" daughter was born early during their family holiday in the county.
The 6lb 1oz girl was delivered by Caesarean section at the Royal Cornwall Hospital in Truro on Tuesday. She had been due next month.
Although the baby has not yet been named, Mr Cameron said: "I think maybe we'll have to have - not that I'll have any say in it, of course - we'll have to have something Cornish in the middle."
He added that his wife was looking through a baby name book "and she'll decide, I'm sure, before long".
The Prime Minister said both the baby and Mrs Cameron were doing "very well".
"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.
His other two children, six-year-old Nancy and four-year-old Arthur Elwen, were "thrilled, bouncing up and down, and dying to say hello to the new baby", he said.
Deputy Prime Minister Nick Clegg, who has been holding the fort in Downing Street during the Camerons' holiday, sent his congratulations to the Prime Minister. He said: "All our best wishes are with David and Samantha at this special time for them and their family."
The Camerons' new baby is only the third born to a serving prime minister since 1849.