"The next thing we knew we were getting into the gear and the baby was coming out. The baby popped out at about 12 o'clock and they seem to be doing fine.
"My job was to make the toast and the tea and things like that. It's absolutely thrilling, really exciting.
"It was a bit of a shock. I always thought it was possible because Samantha's have tended to come a bit early."
When the pregnancy was announced in March, the baby was said to be due in September.
Samantha Cameron, 39, was driven by car to the hospital, arriving at 8:00 am (0700 GMT).
The couple already have a daughter, Nancy, six, and a four-year-old son, Arthur. Their first child, Ivan, who had cerebral palsy and severe epilepsy, died suddenly in February last year, aged six.
They too were all delivered by Caesarean section.
Conservative Party leader Cameron said Nancy and Arthur were "thrilled, bouncing up and down, and dying to say hello to the new baby".
The first child born to a serving British premier in the last 160 years was Tony Blair's son Leo in 2000. Leo Blair was the first since 1849.
Cameron said in March the couple wanted another child.
"We were very keen to have another baby after Ivan died and sometimes it takes a while before the stork pops one down the chimney," he said.
The Camerons started their summer break on August 16 with a few days at Chequers, the prime minister's country retreat northwest of London, before moving on to Cornwall, where they have taken previous family holidays.
The casually-dressed couple had posed for photographs on Sunday at Daymer Bay beach, with a heavily pregnant Samantha wearing a patterned yellow dress and a cardigan.
Deputy Prime Minister Nick Clegg has been in day-to-day charge of the government while Cameron has been away and the Liberal Democrat leader may now find himself "holding the fort" for an extended run.
Clegg said he and his Spanish wife Miriam, who have three young sons, were "absolutely delighted by the fantastic news."
Cameron had been expected to return to work next week, but is now likely to take some paternity leave.
After Cameron became prime minister at the head of a Conservative-Liberal Democrat coalition following the May general election, his family moved from their west London townhouse into 10 Downing Street, the premier's official residence.
However, once some renovation works have been completed, they will follow the example of both Blair and his successor in office, Gordon Brown, by switching to the larger flat above number 11, the finance minister's residence.
No names have been decided for the new baby, but Cameron said his wife was going through a book of Cornish names to have as a middle name.
Bookmakers William Hill said Marnie was the 2/1 favourite for the new baby's name, followed by Enid at 3/1 and Annabel at 4/1.
Margaret, possibly after Conservative former prime minister Margaret Thatcher, was 50/1.