"Her Royal Highness The Duchess of Cambridge was safely delivered of a son at 4.24 pm," Kensington Palace said in a statement just over four hours after she gave birth.
It was in the same hospital wing that the late Princess Diana gave birth to William in 1982.
Royal aides later placed a formal bulletin on a gold easel in the forecourt of Buckingham Palace which said that both Kate and her son were "doing well".
Kate, William and their new baby will remain in hospital overnight, the palace said.
Prince Charles said how "overjoyed" he and his wife are with the arrival of the future King.
"Both my wife and I are overjoyed at the arrival of my first grandchild," he said in a statement.
"It is an incredibly special moment for William and Catherine and we are so thrilled for them on the birth of their baby boy.
"Grandparenthood is a unique moment in anyone's life, as countless kind people have told me in recent months, so I am enormously proud and happy to be a grandfather for the first time and we are eagerly looking forward to seeing the baby in the near future."
Newspapers splashed the arrival over their front pages, with many saying simply: "It's a boy.''
Check out a gallery of headlines from around the world below:
The arrival of a royal baby and a new generation for the British monarchy has been hotly anticipated ever since the couple married in 2011 after a decade-long romance.
The popular couple have been widely credited with revitalising "The Firm", as the British royals are known, following decades of scandal and the death of Diana in a Paris car crash in 1997.
The palace said the boy's name would be revealed later, but he will be known as Prince of Cambridge.
Bookmakers said the favourites are George and James, traditional names for a future monarch - although they strongly backed a female child before the birth.
New Zealand Prime Minister John Key said the birth was "wonderful news".
"The birth of a child is a time of great joy and excitement, and I know they will make excellent parents," he said in a statement.
"New Zealanders remember with fondness the visit of Prince William when he was just a toddler, playing on the lawn of Government House with a Buzzy Bee. It would be a great pleasure to welcome Prince William's son to New Zealand as well."
New Zealand's official gift to the royal couple is a hand-spun, hand-knitted, fine lace shawl, similar to the one gifted to Prince William when he was born.
There was a 21-gun salute at noon today to celebrate the royal birth and nearly 30 landmarks around NZ and more around the world will be lit up blue at sun down to celebrate the arrival of the baby boy.
Monarchy New Zealand came up with the illuminating concept and has managed to get loads of landmarks on board, including the Sky Tower and The Cloud in Auckland; Wellington's Majestic Centre lights; Christchurch Airport; and the corrugated iron dog in Tirau. During Kate's labour the Sky Tower alternated between a pink and blue hue.
US President Barack Obama and his wife Michelle have wished the royal couple "all the happiness and blessings parenthood brings."
He said the new British prince has been born "at a time of promise and opportunity" between Britain and the United States.
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William, also 31, who was at his wife's bedside for the birth, has been on annual leave and will take two weeks' paternity leave from his job as a Royal Air Force search and rescue pilot.
The birth came later than widely expected, adding to the sense of anticipation - as well as the frustrations of hundreds of international media who have waited outside the hospital for three weeks.
Check out a timeline of Kate and William's royal romance here.
Follow the hype on Twitter here with the hashtag #royalbaby:
- www.nzherald.co.nz, AFP
