Endellion! The Cameron baby's Cornish name

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Florence Rose Endellion's third name refers to a village near where the PM and his wife have been holidaying

The couple were said to have finalised the choice of names this afternoon.

Endellion refers to the Cornish village of St Endellion, close to where they are holidaying and which Mr Cameron has visited over the years.

Downing Street said that the couple had received congratulatory messages from the Queen, the Prince of Wales, Gordon and Sarah Brown and French President Nicolas Sarkozy.

Mr Cameron spent the night at the Royal Cornwall Hospital in Truro where his new daughter was delivered by Caesarean section yesterday.

The couple's two elder children, six-year-old Nancy and four-year-old Arthur Elwen, have now visited their sister for the first time.

Christine Rashleigh, director of nursing, midwifery and allied health professions at the hospital, said she hoped Mrs Cameron would get to spend as much time as possible with the new baby before returning to the "cut and thrust of working and supporting the Prime Minister".

She said mothers are kept in for up to 48 hours after giving birth by Caesarean section.

She said: "July and August are really busy times for us. We double our population in Cornwall during the summer due to the holidaymakers and we are always ready for that eventuality.

"A few press people and some special guests are nothing compared to what we normally have to deal with throughout the summertime and the resulting injuries people sustain when they are on holiday."

The Camerons will have to make a personal decision on how they travel home, she said.

"The baby should be absolutely fine. It's Mrs Cameron and her comfort in a five-hour car journey that I would think they would want to consider," she said.

She added: "The nurses will be making sure that Mrs Cameron rests as much as she possibly can and pays attention to her other children as well as her new baby because that's going to be really important for the family to bond as a whole and certainly Mr Cameron will play a very key part in that."

The birth of Florence, who weighed 6lb 1oz, was "a bit of a shock" to the Camerons because Mrs Cameron was not due to give birth until next month.

Mr Cameron said yesterday he was keen to include a reference to the county of her birth in his daughter's name.

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."

They went to the hospital at about 8am yesterday after Mrs Cameron experienced "a couple of days" of contractions.

"We're absolutely thrilled. She is an unbelievably beautiful girl and I'm a very proud dad and both baby and mum seem to be doing very well, so it's really exciting," Mr Cameron said.

The Camerons suffered tragedy last year when their elder son Ivan, who suffered from cerebral palsy and severe epilepsy, died aged six.

Mr Cameron was driven away from the rear of the hospital's maternity unit, the Princess Alexandra Wing, in a black people carrier at about 4.30pm.

The Camerons' new baby is only the third born to a serving prime minister since 1849.

Folding beds are available for fathers who want to stay overnight.

The maternity unit has nine side rooms for mothers who have Caesarean sections or traumatic births.

A hospital spokesman said: "When we had the refurbishment done a couple of years ago that allowed us to increase the number of side rooms.

"It's done on a case-by-case basis and the individual needs of the family.

"We have got a lot of side rooms so the benefit of dads being able to stay is extended to a reasonable number of people.

"A side room is where the mum sleeps with the baby and dads can stay there too if they want to."