Kensington Palace said the baby was born at 1001 GMT and weighs 8 lbs 7oz (3.8 kg). William was present for the birth at St. Mary's Hospital in west London where their other two children, George and Charlotte, were also born, it added.
"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."
The new prince is Queen Elizabeth's sixth great-grandchild and will follow siblings George, 4, two-year-old sister Charlotte, their father William, and grandfather and heir Prince Charles in line to the British throne.
A 2013 change to the law means the boy will not supplant his older sister Charlotte in the order of succession. Previously younger males would take precedence over older female siblings.
"That would, in fact, be the first time that has ever happened," royal historian Hugo Vickers said.
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 favorite 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.
"My warmest congratulations to the Duke and Duchess of Cambridge on the birth of their baby boy," Prime Minister Theresa May said. "I wish them great happiness for the future."
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.
The couple now live at Kensington Palace and last year William gave up his day job as an air ambulance pilot to concentrate full time on royal duties on behalf of his grandmother, having faced some criticism from British newspapers who dubbed him "workshy Wills".
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.
William and Kate's wedding and the birth of Prince George, who is destined to be a future king, generated a worldwide media frenzy. While interest in their third child has been more muted, large numbers of photographers and broadcasters massed outside the hospital, along with some dedicated royal fans.
John Loughrey, 63, from south London, who was wearing a Union Jack onesie and hat and holding a baby doll adorned with a crown, said he had been camped outside the hospital for 15 days.
"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 enamored.
"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.