Congratulatory messages are flooding in from around the world to mark the birth of the Duke and Duchess of Cambridge's son, the third in line to the throne.
Prince William said the couple "could not be happier" following the birth of the boy, who weighed 8lb 6oz and is yet to be named, at 16:24 BST on Monday.
BBC royal correspondent Nicholas Witchell said the expectation is that they will leave hospital on Tuesday.
The birth will also be marked later with a series of gun salutes.
The duke was at the private Lindo Wing of St Mary's Hospital, west London, for the birth - and stayed with Catherine and the baby overnight.
The couple are expected to talk to their medical team before they make any decision to go home.
BBC royal correspondent Peter Hunt said that in the meantime it was unlikely the Queen would visit her great-grandchild in hospital, adding "she can of course see him privately once he leaves".
At Buckingham Palace, royal watchers and tourists are queuing in the rain to catch a glimpse of a bulletin announcing news of the birth, which is being displayed on an easel.
The notice is to be on show for 24 hours, which means it is likely to be removed just after 20:00.
Guardsmen outside the palace brought a celebratory feel to the proceedings ahead of the Changing of the Guard, playing "Congratulations" to the crowd.
The world now awaits the couple's choice of names for their son, with George the bookmakers' favourite, followed by James and Alexander.
Nicholas Witchell said the scale of the international interest was "pretty awesome", and "quite threatening in a sense". He said the couple would be "knocked back" when they step out of the hospital.
The arrival generated headlines and celebrations around the world, and prompted messages of goodwill to flood in:
- New Yorkers were informed of the news when it was tickered in lights at Times Square
- In London, Trafalgar Square was lit blue for a boy, while the BT Tower delivered the message: "It's a boy!"
- At its peak, there were more than 25,000 tweets per minute about the royal baby's birth
- US president Barack Obama said: "We wish them all the happiness and blessings parenthood brings" while Australian PM Kevin Rudd said Australians wished "the royal bub all the best"
- The Archbishop of Canterbury, Most Reverend Justin Welby, greeted the news by saying: "Along with millions here and around the world, I share in their joy at this special time"
- The village of Bucklebury in Berkshire - Catherine's childhood home - was "intensely delighted at the birth", said the BBC's Ben Moore
- A specially-filmed scene will be inserted into Tuesday's edition of EastEnders to mark the occasion
At 14:00 BST, the King's Troop Royal Horse Artillery will stage a 41-gun salute in Green Park, after riding past Buckingham Palace.
At the same time, the Honourable Artillery Company - the City of London's army reserve regiment - will fire a 62-gun salute from Gun Wharf at the Tower of London.
And the church bells of Westminster Abbey, where William and Catherine were married in April 2011, are expected to ring out for three hours from 14:00.
Following the birth announcement, a statement from Kensington Palace said the Royal Family were "delighted".
The Duchess of Cornwall, on a two-day visit to Yorkshire with the Prince of Wales, said it was a "wonderfully uplifting moment for the country" and that mother and baby were "doing well".
The prince said he was "thrilled and very excited", as the couple were congratulated by well-wishers during a walkabout.
"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," said the prince in a statement on Monday night.
Royal doctor Mr Setchell described the new arrival as a "wonderful baby, beautiful baby".
Prime Minister David Cameron, speaking outside Downing Street, hailed the "wonderful news".
The birth of the prince means the monarchy has three generations of heirs to the throne for the first time since 1894.
Nicholas Witchell said the third in line to the throne could expect to be brought up in a "secure and loving environment", shielded from many of the pressures of ordinary life but facing the specific challenges his position will bring.
The baby's future, he said, will be "a lifetime of public curiosity and, in due course, the responsibility of refreshing and taking forward the ancient institution that is the world's best-known hereditary monarchy".