Royal baby: William and Kate present daughter to the world

The Duke and Duchess of Cambridge have introduced their daughter to the world, as they left hospital to take her home to Kensington Palace.

The princess, whose name has yet to be announced, slept in her mother's arms during her first public appearance outside St Mary's Hospital, in London.

The princess - who is fourth in line to the throne - was delivered at 08:34 BST on Saturday after a short labour.

The Duke of Cambridge was present for the birth of the 8lbs 3oz (3.7kg) baby.

The couple emerged from the hospital to crowds of well-wishers and the world's media. They stood on the steps briefly before heading back into the hospital to put the princess in a car seat. Prince William then drove them to Kensington Palace.

They did not speak to the media.

The duchess had been in hospital for just over 12 hours before leaving.

The baby was born less than three hours after the couple arrived at the Lindo Wing, at St Mary's Hospital.

Outside the Lindo Wing: BBC royal correspondent Daniela Relph

It all happened rather quickly in the end.

Within 12 hours of arriving at the hospital in the early stages of labour, the duchess had left St Mary's with her new baby in her arms and her husband at her side.

That moment on the Lindo Wing steps - the royal couple and their daughter - will be one of the defining images of the year.

Wrapped in a white shawl for her first photocall, the baby princess appeared to sleep all the way through it.

There'll be no name announced this evening. That may take a few days.

The Cambridges are now back at Kensington Palace for their first night together as a family of four.

The family will stay for a few days at Kensington Palace before decamping to their country home, Anmer Hall on the Sandringham estate in Norfolk.

The duchess had been admitted to the private Lindo Wing at St Mary's Hospital in Paddington, west London, at 06:00 BST.

The birth announcement came about 11:00 BST, and the couple left the hospital with their daughter shortly after 18:00 BST.

While the birth was announced on Twitter, there was still room for tradition with an easel announcing the birth placed in the courtyard of Buckingham Palace. The birth was also proclaimed by a town crier who was not on official duty but certainly entertained the crowds outside the hospital.

Prince William left the hospital briefly in the afternoon to pick up his 21-month-old son and told the waiting crowds the couple were "very happy".

Prince George appeared to be the only visitor and he was later taken home ahead of his parents and little sister.

In a statement, Kensington Palace said: "Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have all received.

"They would also like to thank everyone for their warm wishes."

Gun salutes will be fired in Hyde Park and at the Tower of London on Monday to mark the birth.

Messages of congratulations poured in throughout the day for the couple and their second child, including ones from Prime Minister David Cameron and the Archbishop of Canterbury Justin Welby.

The Prince of Wales, who had earlier said he was hoping for a grand-daughter, and the Duchess of Cornwall said they were "absolutely delighted".

In addition to the die-hard royal fans, who had been camping outside the hospital in Paddington for days, hundreds of well-wishers descended when the news first broke that the duchess had been admitted to hospital.

The next news to be announced is likely to be the name of the princess, who is the Queen's fifth great-grandchild.

Alice is the favourite with several bookmakers, although Charlotte, Elizabeth and Victoria have also been popular with punters.

Prince George's name was announced two days after his birth in July 2013, which is a relatively quick turnaround compared with previous royal babies.

It was seven days before the name of a newborn Prince William was announced in 1982, and there was a wait of a month following Prince Charles's birth in 1948.