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3 May 2015, 06:01 | Updated: 3 May 2015, 06:02
The Cambridges will stay for a few days at Kensington Palace before decamping to their new country bolthole, Anmer Hall on the Sandringham estate in Norfolk.
After a few days at Kensington Palace in London with the Princess, they'll make their way to Anmer Hall, a ten-bedroom mansion on the Sandringham Estate.
It was was a wedding gift from the Queen.
Meanwhile it's thought the name of the Duke and Duchess of Cambridge's baby daughter could be announced by her parents today.
William and Kate may tell the world what they will call their second child who was born yesterday and introduced to the world at a photocall outside hospital.
The Cambridges announced the name of their son George the day after the couple left hospital in July 2013, so may do the same for their daughter.
The Duke and Duchess appeared overjoyed when they left hospital on Saturday with their baby princess who was born at 8.34am, weighing 8lbs 3oz.
Despite screams from royal fans and the clicks of dozens of camera shutters the infant did not stir and slept serenely on, wrapped in a shawl and wearing a bonnet.
Proud father William spoke of his joy following the birth earlier that day, telling the waiting press he was ``very happy''.
He had been at his wife's side in the delivery room at the private Lindo Wing of St Mary's Hospital in Paddington.
The Duke's father the Prince of Wales - who said he wanted his second grandchild to be a girl - and Duchess of Cornwall were left ``absolutely delighted'' by the news, Clarence House has said.
William's uncle Earl Spencer also reacted with joy to the announcement, saying: ''It's wonderful news - we are all thrilled for all four of them.''
The Queen, who attended a military event at Richmond Castle, in North Yorkshire, was dressed appropriately in pink on the day her fifth great-grandchild, who was overdue, was born.
Prince George, now 21 months old, was picked up by his father from the family's Kensington Palace apartment and taken to the private maternity unit to meet his baby sister.
It was the first time he had been seen at a public event in the UK since he left hospital after his birth, but after waving to crowds when he first arrived he left privately by a rear entrance.
Among the bookies the names Alice and Charlotte are vying for top spot - with tens of thousands wagered by punters - but the name Olivia is also attracting bets.
William Hill spokesman Rupert Adams said: ``It has been mayhem with Charlotte and Alice very much the favourites. This late move for Olivia has taken us by surprise and it has to be a strong contender.''
He said Charlotte was once again the favourite at 3/1, with Alice 7/2, while Olivia had been the big mover cut from 33/1 to 5/1.
Bookmaker Coral has also slashed the odds on the new princess being called Olivia.
The firm was forced to cut the odds on Olivia to 10/1 from 50/1, Charlotte is 5/2, while the odds on Alice, the favourite, have drifted out to 7/4 from 5/4.
Coral's Nicola McGeady said: ``The gamble on Olivia was unexpected, but the rush of bets in favour of it suggests that it is very much in the running to be the princess' name.''
The Cambridges' daughter, who is fourth in line to the throne and will be affectionately known as a spare to the heir, looked very similar to her great-grandmother the Queen when she was a newborn.
The birth was proclaimed on Twitter, by the age-old tradition of a bulletin displayed in Buckingham Palace's forecourt and by a town crier who entertained the crowds who had gathered outside St Mary's Hospital.
Palace footmen George Oates, 26, from North Yorkshire, and Simon Garnett, 30, from Cumbria were given the job of placing the proclamation on an ornate gold easel which was sited close to the palace railings.
Very quickly a long queue formed as people waited patiently to get a glimpse of the brief bulletin signed by the senior medical team, led by consultant obstetrician Guy Thorpe-Beeston, surgeon-gynaecologist to the household, who delivered the royal baby.
There were also words of congratulations from all the political party leaders who broke briefly from campaigning for the General Election.
Gun salutes will be fired in Hyde Park and at the Tower of London on Bank Holiday Monday afternoon to mark the birth.
The Cambridges are expected to remain at their Kensington Palace apartment for another day before they travel to their country retreat Anmer Hall in Norfolk.
While in London they are likely to be visited by proud grandparents Michael and Carole Middleton and possibly members of the Royal Family.