More celebrations are expected Tuesday UK time, including gun salutes by royal artillery companies to honor the birth and the ringing of bells at London's Westminster Abbey.
Kate, William and their son spent the night at the private Lindo Wing of St. Mary's Hospital. Officials have not said when the family will leave.
Tourists and well-wishers lined up outside Buckingham Palace gates Tuesday to take pictures of the golden easel on which, in keeping with royal tradition, the birth announcement was displayed.
"This was a great event yet again our royal family is bringing everyone together," said 27-year-old David Wills, who took a two-mile detour on his run to work to pass the palace. "I kind of feel as though I am seeing part of history here today."
The baby isn't even a day old and may not be named for days or even weeks but he already has a building dedicated to him.
Australian Prime Minister Kevin Rudd said an enclosure at Sydney's Taronga Park Zoo would be named after the prince as part of a gift from Australia. The government would donate 10,000 Australian dollars on the young prince's behalf toward a research project at the zoo to save the endangered bilby, a rabbit-like marsupial whose numbers are dwindling in the wild. The prince's name when known will be added to the bilby enclosure.
"I don't know if the royal family would need this, but we'll probably give them a free pass to Taronga Park Zoo as well," Rudd said.
British media joined in the celebration.
"It's a Boy!" was splashed across many U.K. front pages, while Britain's top-selling The Sun newspaper temporarily changed its name to "The Son" in honor of the tiny monarch-in-waiting.
"Regal has Landed," the paper cried.
The Mirror, in an allusion to the fact that news of the baby's birth first broke online, spoke of: "The Twitter of Tiny Feet."
So far, it's the palace which has largely set the pace of the wall-to-wall media coverage, using social media to keep followers abreast of the news.
Check out a gallery of headlines from around the world below:
New additions to the royal family typically set off a tug-of-war between the royal household and the press. The feverish media excitement ahead of Kate's engagement to William led to a warning from Britain's press watchdog, and the hunger for news about the third-in-line to the British throne is likely to be similarly intense.
William, whose mother Diana died in a Paris car crash in 1997 while being hounded by paparazzi, was likely to insist on a measure of privacy, former royal press secretary Charles Anson told the BBC.
"I think Prince William and the duchess will be very keen to protect their young child from too much attention," he said.
On the other hand, he said that "people want more news about this royal baby."
He said the palace would have to find some way to handle that, predicting "a bit of negotiation, a bit of arrangement with the media to get it right."
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."
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, AP
