"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.
"Her Royal Highness and her child are both doing well and will remain in hospital overnight."
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."
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.
Kate and William, who spent the weekend at Kensington Palace, travelled to the hospital without a police escort just before 5pm Monday, NZ time, their spokesman said.
The Duchess gave birth to the baby at the private Lindo Wing of St Mary's Hospital in Paddington, west London, where William was born in 1982.
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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's name remained a mystery for an entire month.
The new baby boy will become third-in-line to the UK throne, behind his grandfather Prince Charles and father Prince William.
