Couple delay announcement of the birth for four hours because they wanted to spend time with baby first.
Prince Charles celebrated the birth of his grandson with bubbles, and his wife Camilla, the Duchess of Cornwall, said the news of William and Kate's new prince was "wonderfully uplifting".
As William and Kate prepared to introduce their son to the world, Charles and Camilla were in a buoyant mood as they visited Yorkshire early today (NZT).
"I think it's wonderful news," Camilla told the BBC. "I think mother, son and father are all well. And I think it's a wonderfully uplifting moment for the country.
"It's very exciting and it's wonderful for the grandfather - he's brilliant with children."
During the Yorkshire visit, one man, Robert Barrett, reportedly greeted the Prince of Wales, with the words, "Morning Granddad".
Charles "chuckled" in reply, reported the Guardian.
The baby's name was a hot topic for the pair, but Charles said the name wasn't clear yet.
The Guardian reported that another Yorkshire resident, Alec Dale, told the prince: "We popped a bottle of bubbly last night at our house. I hope you did too."
Charles replied: "Yes. But just a little bit."
William, Kate and their new son were still at St Mary's Hospital in Paddington early today - and not expected to leave until breakfast (NZT) at the earliest and possibly not until tonight.
But they issued a statement, saying: "We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received.
"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."
The Duke of Cambridge, who was present for the birth, said in an earlier statement: "We could not be happier."
News that the Duchess of Cambridge had given birth to a baby boy sparked celebrations in Britain and around the world yesterday. Thousands of wellwishers gathered outside Buckingham Palace last night (NZT), to watch the changing of the guard as a military band played the song Congratulations.
William phoned the Queen to give her the news. He then called the Prince of Wales, Prince Harry and the Middleton family, who were able to speak to the Duchess briefly.
The couple decided to delay the announcement of the birth for four hours because "they wanted to spend a bit of time with their baby first", an aide said.
The newest addition to the royal family who becomes third in the line to the throne, ahead of Prince Harry, was born naturally, it was said.
"The whole country will celebrate," Prime Minister David Cameron said. Of Prince William and Kate, he said: "They'll make wonderful parents."
The Queen was not expected at the hospital, according to the Daily Telegraph's Gordon Rayner. He tweeted last night she had a "busy day".
Prime Minister John Key said it was "wonderful" news. He hoped the male heir would one day play on the lawn of Government House as his father did in 1983 when he was a child.
Name expected to follow tradition
The Duke and Duchess of Cambridge are expected to pick a traditional royal name for their newborn.
Royal infants are mostly given historical names, passed down through the generations, and Debrett's Peerage and Baronetage editor Charles Kidd predicts William and Kate will follow that convention.
"I do think that Prince William is quite a traditionalist," Mr Kidd said.
"I wouldn't be surprised if they use a name familiar in the royal family."
He suggested George could be their name of choice.
Here are the latest odds for the royal infant's name:
1) James (2/1)
2) George (5/1)
3) Henry (5/1)
4) Arthur (20/1)
5) Louis (25/1)
6) Albert (40/1)
7) Alexander (40/1)
8) Benjamin (40/1)
9) David (40/1)
10) Philip (40/1)
Source: William Hill and Coral