The couple, their firstborn Prince George and the new arrival were expected to spend the next few days relaxing together at their Kensington Palace apartment, a day after Kate gave birth to the baby girl, who is fourth in line to the throne.
A beaming Duke and Duchess of Cambridge left hospital on Saturday carrying their newborn.
They were cheered by hundreds of well-wishers who had gathered outside St Mary’s Hospital in central London along with the massed ranks of photographers from the world’s media.
The sleeping baby was wrapped in a woollen shawl as a relaxed-looking Kate, 33, cradled her in her arms in front of the hospital’s private Lindo Wing.
William, 32, then drove his wife and daughter the short distance to Kensington Palace after carrying the baby out in a car seat.
First visitors were likely to include Kate’s parents, Michael and Carole Middleton and William’s father Prince Charles and his wife Camilla.
William’s grandparents Queen Elizabeth II and her husband Prince Philip will also be eager for a first glimpse of their fifth great-grandchild.
Charlotte is new bookies’ favourite
Britain is eagerly awaiting the announcement of the new princess’s Christian names.
Charlotte became the new favourite with the bookmakers on Sunday, overtaking long-time favourite Alice.
Charlotte was trading at odds of around 3/1, followed by Alice at 4/1, Victoria at 5/1, Olivia at 6/1 and Elizabeth at 7/1.
The royals could announce the name as soon as Sunday, although the world waited two days before learning George’s name.
The newborn is the first princess born into the British royal family since Princess Eugenie, daughter of the queen’s second son Prince Andrew, 25 years ago.
British Prime Minister David Cameron led the congratulations.
“This is wonderful news and together with the whole of the country I want to give the warmest possible welcome to the new princess,” he told reporters.
“Kate and William are a remarkable couple. They’re loving, wonderful parents and this will be such a precious time.
“The whole country celebrates.”
US President Barack Obama and his wife Michelle spoke of their delight at the birth.
“On behalf of the American people, we wish the duke and duchess and their son George much joy and happiness,” the White House said in a statement.
London’s landmarks including Tower Bridge and Trafalgar Square were lit up in pink in celebration as was the Hilton London Paddington Hotel near the Lindo Wing.
Britain’s Sunday newspapers, with one exception, carried big front-page pictures of the new baby, with many running souvenir pullouts.
“Sleeping cutie,” said both The Sun and the Sunday Mirror, with a close-up of the baby’s face.
The birth is “a reminder of the things that really matter in life”, the Sunday Telegraph said.
Chinese fortune tellers meanwhile predicted that the new princess would be strong-willed and innovative, a free spirit who would re-energise the royal family.
Rare glimpse of Prince George
Kate gave birth within three hours of being admitted to hospital.
In a mix of old and new, the birth was announced in a tweet from Kensington Palace and on a proclamation placed on a gilded easel in front of Buckingham Palace, where hundreds of onlookers gathered to witness the historic moment.
William was present for the birth and the princess weighed in at eight pounds and three ounces (3.7 kilogrammes).
William then left to bring George to the clinic to meet his little sister.
It was the first time since his own birth at the hospital in July 2013 that George had been seen at a public event in Britain as his parents have shielded him from the media.
The new princess sits fourth in line to the throne after Queen Elizabeth’s eldest son Charles, 66, his eldest son William and his eldest child George.
She is the first major royal who cannot be overtaken in the line of succession by any future younger brothers following changes to the monarchy’s rules to remove gender bias.
The royal couple are planning to spend the next few days at Kensington Palace.
They will then travel to Anmer Hall, their secluded 10-bedroom country mansion on Queen Elizabeth’s privately-owned Sandringham estate in Norfolk, eastern England.
Gun salutes will be fired across London on Monday to mark the birth. – AFP