The Camerons have chosen the name, Florence Rose Endellion for their baby daughter.
Mr Cameron did say there would be a Cornish link in the baby's name to remind them of their special time in Cornwall.
Flowers and good will messages have been arriving at the Royal Cornwall Hospital following the birth of the Prime Minister's daughter.
Speaking on BBC Radio Cornwall, Mr Cameron said: "We are absolutely thrilled. She is an unbelievably beautiful girl.
"I am a very proud Dad. Both baby and Mum seem to be doing very well.
"The other two children are bouncing up and down and dying to say hello to the new baby."
Samantha Cameron gave birth at lunchtime on Tuesday 24 August, while the family were still on holiday in Cornwall.
As soon as the news was announced, gifts were soon arriving at the hospital for the new arrival.
The Prime Minister says he's a very proud Dad
Katrina Scarlett, from Camborne, brought her own gift.
She says: "The kids wanted me to bring a fudge cake up, and it's still warm. I passed it on to Mr Cameron's entourage.
"I know what it's like when you get a new child, you like to have a bit of luxury for yourself."
Mr Cameron praised the Royal Cornwall Hospital at Treliske, where the baby girl was born at lunchtime on Wednesday 24 August.
He said: "Wonderful hospital, they were all so friendly. Everyone put us completely at our ease. They were all absolutely wonderful. Lovely to be in Truro and to have the baby here."
Staff at the hospital were delighted everything went well with the birth of the Prime Minister's daughter. The hospital's Trust thanked the Prime Minister for, what it called, his kind and generous words about the care his wife and baby had received.
Maternity matron Helen Ross-McGill says every birth is thrilling
Helen Ross-McGill is the Matron at the Princess Alexandra Maternity wing.
She says: "As a midwife I am thrilled by every baby and every positive outcome. We're always happy for all Mums."
Samantha Cameron gave birth to a daughter yesterday by emergency Caesarean section.
The baby was several weeks early, weighing in at six pounds and one ounce.
Helen Ross-McGill says there's usually a number of reasons why this procedure would be carried out.
"It could be that the labour doesn't progress normally, or that the baby is in distress, or there are some problems with the mother."
Father Nicholas Leviseur, a chaplain visiting the hospital, said it was a wonderful occasion:
"It's a very happy day, it's a terrific day. It's great for Cornwall, but most importantly great for that family. A new child, new life, new hope."
The latest addition to the Cameron family is only the second baby to be born to a serving Prime Minister in more than 150 years.
The last being Tony and Cherie Blair with their son Leo who was born in May 2000.
Mr Cameron did say there would be a Cornish link in the baby's name - and they have chosen the name 'Endellion' as one of the middle names.
BBC Radio Cornwall has been flooded with Cornish name suggestions including Kerenza, Tamara, Merryn, Demelza, Megan among others.
Cornwall Council said it's delighted that the Camerons links with the county will be stronger than ever.
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