LONDON (REUTERS) - Prince William and his wife Kate gave the world its first glimpse of their new baby on Tuesday when they left hospital to a barrage of camera flashes with the boy who is third in line to the British throne.

Cheered by well-wishers, the couple waved and took turns to hold their son, wrapped tightly in a white blanket with only the top of his head and his hands visible.

"We are still working on a name, so we will have that as soon as we can," Prince William told reporters, adding that he could not be happier.

"He's got her looks, thankfully," he added, glancing at his wife.

Kate, wearing a pale blue dress with white polka dots, brushed aside the compliment, saying "No, no, I'm not sure about that."

"It's such a special time," she added as onlookers hung out of the windows of surrounding buildings for a better view. "Any parent will probably know what this feeling is like."

The couple followed tradition by giving the public the first sight of the baby on the steps of the hospital as they left - just as William's father Prince Charles and late mother, Princess Diana, did with him.

A relaxed William laughed as he confirmed he had already changed his first nappy (diaper) and joked that he would remind his son "of his tardiness when he is a bit older", because the infant had kept them, and the public, waiting longer than expected.

"He's got a good pair of lungs on him, that's for sure. He's a big boy, he's quite heavy," Prince William added.

WEEKS OF ANTICIPATION

Kate, 31, gave birth to the couple's first child on Monday afternoon, ending weeks of feverish anticipation about the arrival, and all three spent the night in the hospital - St Mary's in west London.

Their baby cried and wiggled his fingers as William carried him in a car seat to a Range Rover before driving his family to nearby Kensington Palace, his childhood home.

Kate's parents, Carole and Michael Middleton, were the first visitors to see the couple and their new child. Asked if she would reveal the name or had made any suggestions, Kate's mother said: "Absolutely not." Apart from the media, the event attracted hundreds of well-wishers from the public.

"We're here to witness history, where a future monarch has been born," said Ms Maria Scott, a housewife who had camped outside the hospital since Saturday.

The baby is third in line to the throne after grandfather Charles and William, 31, and pushes the fun-loving Prince Harry, William's brother, into fourth place.

"I'm thrilled," Prince Charles said. His second wife, Camilla, spoke of a "wonderfully uplifting moment for the country", saying Prince Charles would make a "brilliant" grandfather.

Congratulations flooded in from around the world after the birth, which was followed moment-by-moment by global media.

WRAPPED IN THE FLAG

It fuels a new wave of popularity for the House of Windsor inspired by the younger royals, William and Harry.

Support for the royal family dipped after their mother Diana's death in a car crash in Paris in 1997, a year after her divorce from Charles, as the family's handling of the aftermath prompted accusations that they were out of touch.

But last year's celebrations of Queen Elizabeth's 60th year on the throne showed support for the monarchy at a record high.

Hordes of TV crews, photographers and royal fans wrapped in British flags camped outside the hospital for the first glimpse of the baby, to be called the Prince of Cambridge.

To mark the birth, the Royal Artillery fired a 41-gun salute at London's Green Park and 62 rounds were fired at the Tower of London. The bells of Westminster Abbey, where the couple married in April 2011, rang for three hours.

Prince William and Kate, who met when they were students at university in Scotland a decade ago, are officially known as the Duke and Duchess of Cambridge.

The couple, who have been living in a cottage in Wales where Prince William is based as a Royal Air Force helicopter pilot, will eventually live with their baby at Apartment 1A in Kensington Palace.

Royal experts said the baby would now be taken out of the public glare.

"Having a baby is a very private moment, and they are a private couple, so the next time we see the baby will be the official photo, and that could be weeks," said Joe Little, managing editor of Majesty magazine.