The baby boy was born at 11:01 BST, weighing 8lbs 7oz
Mother and baby are "doing well", Kensington Palace says
Prince William was present for the birth at the Lindo Wing of St Mary's Hospital, London
The new baby is fifth in line to the throne
His older brother Prince George and sister Princess Charlotte have visited him in hospital
The baby's name is yet to be announced...
Image source, PAAnd finally, Kensington Palace says Prince William will be concentrating on his family for the next couple of weeks.
His son's name has not yet been announced, but when asked, the duke said: "You'll find out soon enough."
That's all for today.
The Duke and Duchess of Cambridge aren't the only ones welcoming a new baby today.
Read the stories of Louis, Shaan and Isabell.
Image source, Jessica, Steph and JasA statement from Kensington Palace said: "Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have received.
"They would also like to thank everyone for their warm wishes."
Here's one of the first close-up pictures of the new prince, cradled by a beaming Duchess of Cambridge.
Image source, PAAfter that first appearance in the Duchess's arms, the new prince re-emerged in a car seat and the family is now on their way home to Kensington Palace.
As they left, Prince William said they were "very happy".
And here's the moment when the new addition to the royal family made his first public appearance.
The Duke and Duchess have introduced their baby boy to the world as they emerged onto the steps of the hospital's Lindo Wing.
The couple waved to the media and paused on the steps for photographs as crowds cheered.
The Palace added: "Prince George and Princess Charlotte have now returned to Kensington Palace.
"Their Royal Highnesses would like to thank all staff at the hospital for the care and treatment they have received.
"They would also like to thank everyone for their warm wishes."
Kensington Palace has announced that the Duke and Duchess will leave hospital this evening and return home with their new son.
Could the royal baby boost the economy?
Views are mixed. But one firm is so confident it's willing to put a figure on it.
Independent brand valuation and strategy consultancy Brand Finance estimates the boy will add £50m to the UK economy before he reaches his first birthday.
Chief executive David Haigh says: "The birth of the new prince is also a tremendous marketing opportunity for British producers and retailers of baby products who can reference the royal baby in their promotional campaigns."
Prince William has returned with his older children, having successfully got them out of Kensington Palace and into the car.
Prince George, four, made the same journey when Princess Charlotte was born just under three years ago. At the time many joked about Prince William's ability to buckle his son into a car seat.
Charlotte waved to the crowd and waiting media as the family disappeared behind the famous door of the Lindo Wing to meet their baby brother.
Historian Professor Kate Williams has just about squeezed in alongside the photographers and cameramen from across the globe, waiting with baited breath to get a glimpse of the new fifth in line to the throne.
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Prince William will be back at the hospital soon...
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Prince William has left the Lindo Wing to pick up his two older children and bring them to meet their new baby brother.
"Back in a minute!" he said to the waiting media.
Prince George, four, has been at school as normal today, with two-year old Princess Charlotte spending the day at home.
It is unclear whether we will see the whole family together, but the older children's arrival will echo scenes from Princess Charlotte's birth, when the Duke left the hospital and returned with Prince George, who gave the crowd a shy wave.
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By Jonny Dymond, BBC royal correspondent
Image source, AFPAfter some quieter times, the Royal Family is back in the news - and there is is a noticeable sense of renewal; new faces, with Meghan Markle, Prince Harry's fiancee, making her first public appearances, and now another child, the as-yet unnamed third child of Duke and Duchess of Cambridge.
It's always gone like this - waves of interest followed by periods of relative quiet. But there has been a sense of renewal in the past few months.
Of course there has been concern over the health of the Duke of Edinburgh, and interest in the Queen, with her decades of service, never falters.
But the attention of a new group, both in Britain and around the world, has been caught by this birth and then next month the wedding of Prince Harry and Ms Markle.
From the darkest days of the World War Two to the Instagram generation, the Royal Family now connects them all.
More than 93,000 people have "liked" Kensington Palace's tweet announcing the birth of the new prince.
A further 32,000 have retweeted it as the #Royal baby has been trending on Twitter.
Almost 3,000 have replied to the tweet with the majority offering congratulations
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