"We know it has been a very busy period for the hospital and we would like to thank everyone — staff, patients and visitors — for their understanding during this time," he said in a statement.
The couple's Kensington Palace office said Kate, the Duchess of Cambridge, had given birth to the 8 pound, 6 ounce (3.8 kilogram) baby boy at 4:24 p.m. Monday, triggering an impromptu party outside Buckingham Palace and in front of the hospital's private Lindo Wing.
The palace said Tuesday that "mother, son and father are all doing well this morning."
The new family was expected to remain in the hospital until Tuesday evening or Wednesday morning.
In the meantime the infant's appearance — and his name — remain a royal mystery.
Tourists and well-wishers flocked to Buckingham Palace on Tuesday, lining up outside the gates 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."
A band of scarlet-clad guardsman at the palace delighted onlookers with a rendition of the song "Congratulations."
Other celebrations Tuesday included gun salutes by royal artillery companies to honor the birth and the ringing of bells at London's Westminster Abbey.
Halfway around the world, royalist group Monarchy New Zealand said it had organized a national light show, with 40 buildings across the islands lit up in blue to commemorate the royal birth, including Sky Tower in Auckland, the airport in Christchurch, and Larnach Castle in the South Island city of Dunedin. A similar lighting ceremony took place in Canada; Peace Tower and Parliament buildings in the capital, Ottawa, were bathed in blue light, as was CN Tower in Toronto.
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 ($9,300) 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.
British media joined in the celebration, with many newspapers coming out with souvenir editions.
"It's a Boy!" was splashed across many 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.
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."
Beyond the newsstands, the birth of the royal baby brought welcome tidings in a country where polls show the monarchy is as popular as any time in recent history. Most Britons are thrilled to have a fresh royal heir, and in the Yorkshire village of Bugthorpe — which Prince Charles was visiting as part of a tour through northern England — the baby was on everyone's lips.
"Morning Granddad," said local resident Robert Barrett, which drew a chuckle from the prince.
Back in London, there was a healthy interest in the baby's name, combined with a note of concern for his future.
"I hope the child is given the opportunity to have a normal childhood," said Julie Warren, a 70-year-old retired schoolteacher waiting for her grandson outside one of the capital's subway station.
Others expressed less interest.
"It's a baby, nothing else," said Tom Ashton, a 42-year-old exterminator on his way to work. "It's not going to mean anything to my life."
The skeptics notwithstanding, new additions to the royal family typically set off a tug-of-war between the royal household and the media as the public clamors for news. The feverish media excitement ahead of Kate's engagement to William led to a warning from Britain's press watchdog, and the appetite for footage, pictures, and other details about the third-in-line to the British throne is likely to be similarly intense.
So far, it's the palace which has largely set the pace of the wall-to-wall coverage, using social media to keep followers abreast of the news.
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.
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."