"The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news," Kensington Palace said.
"Her Royal Highness and her child are both doing well."
A 2013 change to the law means the boy will not supplant his older sister Charlotte in the line of succession. Before that change younger males would take precedence over older female siblings.
William's younger brother Harry, who marries U.S. actress Meghan Markle next month, falls to sixth on the list.
Kensington Palace said the name of the new royal would be announced in due course. Bookmakers have made Albert, Arthur, Jack and Fred the favourite names for a boy.
In keeping with tradition, the official announcement of the birth will be made by placing a notice on an easel in the forecourt of the queen's London home, Buckingham Palace.
William and Kate met as students at St Andrews University in Scotland. They got married in a pageant-packed ceremony at Westminster Abbey in April 2011 and have since become well-known across the global stage.
He has been determined to play a hands-on role in bringing up his children while also protecting their privacy, an issue close to his heart after the death of his mother, Princess Diana, in a 1997 Paris car crash as her limousine sped away from chasing paparazzi.
"I think he takes on a lot more engagements than he used to," said royal biographer Claudia Joseph.
"But compared to members of the general public who have full-time jobs, he still has some time off to look after the children, to give them their baths, to put them to bed and I think he'll continue to do that," Joseph said.
The British royals are rarely out of the media glare, but the baby arrives at a time when the Windsors have particularly been in the limelight.
On Saturday William joined the queen and other senior royals, with the exception of Kate, as the monarch celebrated her 92nd birthday at a televised concert.
The queen also played a prominent role at the Commonwealth Heads of Government meeting in London last week, where it was announced that Charles would succeed her as head of the network of mainly former British colonies.
"We've been camped here outside the hospital for 15 days," said John Loughrey, 63, from south London, who was wearing a Union Jack onesie and hat and holding a baby doll adorned with a crown that said 'feed me mummy' every few seconds.
"We've been coming out doing this for 21 years. We were here for George and Charlotte and we'll be there for the royal wedding too," he said, referring to Harry and Meghan Markle.
However, not everyone in Britain is so enamoured.
"The arrival of a new prince or princess highlights the unfairness of the hereditary principle - for the millions of children told they'll never be good enough and for the baby whose life has been predetermined," anti-monarchist group Republic said.
"The absurd media speculation ignores these serious questions while showing us the kind of intrusion the royal baby can expect for the rest of its life."
The birth will give Kate's own family another cause for celebration after it was reported by British media on Sunday that her sister Pippa was pregnant with her first child.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)