It's a boy! Prince William and Kate "couldn't be happier", the Queen is "delighted", and Prince Charles is looking forward to "grandparenthood".
The 3.8-kilogram (8lb 6oz) heir to the throne, whose name is yet to be announced but who will have the title the Prince of Cambridge, put an end to the waiting of royal-watchers around the world with his arrival at 4.24pm on Monday (3.24am today NZT).
Prince William, who was present for the birth, released a statement saying he and the Duchess of Cambridge "could not be happier".
William phoned his grandmother the queen to give her the news, and also contacted his father Prince Charles and brother Prince Harry.
The Queen, the Duke of Edinburgh, the Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families were "delighted with the news".
"Her Royal Highness and her child are both doing well and will remain in hospital overnight," a statement from Kensington Palace said.
Charles, Prince of Wales said he and his wife Camilla, the Duchess of Cornwall, were "overjoyed at the arrival of my first grandchild".
"It is an incredibly special moment for William and Catherine and we are so thrilled for them on the birth of their baby boy," Prince Charles said.
"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."
The Royal doctor said the child was a "wonderful baby, beautiful baby."
British Prime Minister David Cameron said: "I'm delighted for the Duke and Duchess now their son has been born. The whole country will celebrate."
"I think New Zealanders will be celebrating with the happy couple and it's great to see the birth of a future king of New Zealand so we wish them all the very best," Mr Key said.
"It will be an exciting time for them." offered his congratulations.
It is a historic moment for the British monarchy - the baby will be third in line for the British throne, after Prince Charles and William, leapfrogging Harry, and should eventually become king.
As the birth of the queen's third great-grandchild was announced, a loud cheer went up from the well-wishers and media gathered outside St Mary's Hospital in west London, where William was also born to the late Princess Diana in 1982.
Within minutes, messages of congratulations began flooding in, while crowds gathered outside the queen's London residence Buckingham Palace where an official notice was placed on a gold-coloured easel at the main gates.
US President Barack Obama was one of the first world leaders to welcome the birth.
"Michelle and I are so pleased to congratulate The Duke and Duchess of Cambridge on the joyous occasion of the birth of their first child," he said. "We wish them all the happiness and blessings parenthood brings."
Outside Buckingham Palace, there was a party atmosphere with well-wishers laying flowers and teddy bears, singing God Save the Queen and Happy Birthday, and children waving flags.
"The build up to the birth has been so big I'm just happy it's finally come. I'm pleased it's a boy, you always want a boy really," said Alice Durrans, who rushed from a nearby restaurant after hearing the news.
Deborah Beeson, a banker from the United States, was ecstatic.
"It's wonderful. I got chills. I cried," she said. "You know America loves Kate. She's just beautiful, she has dignity."
There will be a 41-gun salute at London's Hyde Park and 62 rounds fired at the Tower of London tonight to herald news of the birth.
The baby arrives at a time when the royal family is riding a wave of popularity. An Ipsos Mori poll last week showed 77 percent of Britons were in favour of remaining a monarchy over a republic, close to its best-ever level of support.
LONG WAIT OVER
It could be some time before the baby's name is made public. When William was born, a week passed before his name was announced. Charles' name remained a mystery for a month.
The birth of a new heir to the throne had been breathlessly anticipated by many Britons since William and Kate Middleton wed on April 29, 2011.
Reflecting the fact that not everyone was so excited, however. The Guardian added a "republican" button to its website so that those who were not in favour of the monarchy could get a baby-free version of the paper.
Despite a rough start to the pregnancy, when she was hospitalised for acute morning sickness, the 31-year-old duchess made a number of public appearances that stopped only near the end of her term.