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...
Kensington Palace said the name of the baby would be announced "in due course".
A formal notice of the birth will also be posted on the forecourt of Buckingham Palace.
The arrival of the prince has had bookmakers outside the Lindo Wing scrabbling for new odds on his name.
Need a refresher on who the new prince's relatives are?
He is the Queen's sixth great-grandchild and arrived just days after her 92nd birthday.
The new prince has arrived just in time for his uncle's wedding - although he is unlikely to be among the 800 guests.
Prince Harry and Meghan Markle are due to tie the knot in Windsor in 26 days - on 19 May. The Duchess of Cambridge is expected to be attending but, with her baby son only a few weeks old, she is not likely to stay for the entire celebration which will stretch into the evening and probably the early hours.
Image source, PA WIREHer attendance is likely to depend on how she is feeling and how the baby is feeding.
The new prince's brother George and sister Charlotte are tipped to have roles in the wedding.
It is likely if Catherine does attend, the baby will stay with the Cambridges' nanny Maria Teresa Turrion Borrallo.
Alongside Theresa May, who has already tweeted, external her congratulations, other politicians have sent well wishes.
Sadiq Khan, the mayor of London, said: "On behalf of Londoners, I want to send our warmest congratulations to the Duke & Duchess of Cambridge on the birth of their baby boy."
Other tweets came from Brandon Lewis, chair of the Conservative Party, and shadow education secretary Angela Rayner.
With the news of the arrival announced, the next thing those outside St Mary's Hospital is waiting for is a glimpse of the new prince (apart from the name of course).
This was the world's first sighting of Prince George in July 2013:
Image source, Getty ImagesAnd here they are with his sister Princess Charlotte in May 2015:
Image source, ReutersThe new prince is the first royal boy who will not be ahead of his older sister in the line of succession to the throne.
Until 2012, ancient royal succession rules meant male heirs always overtook girls, but the Queen changed the rules ahead of Prince George's birth.
However, the new prince's arrival means his uncle Prince Harry has been bumped down to sixth-in-line to the throne.
The Royal Navy has sent its congratulations to the Duke and Duchess of Cambridge.
Sailors and Royal Marines aboard the HMS Albion, which is on operations in the Asia Pacific Operation, lined up for this photo opportunity.
Image source, Royal NavyThe BT Tower, one of London's famous landmarks, has been lit up in blue to welcome the new baby.
White lettering declaring "It's a boy!" on a blue background is displayed at the top of the tower.
The birth was announced with the traditional appearance of an unofficial town crier on the steps of the Lindo Wing.
His cry of "God save the Queen" was met with cheers from the waiting crowds.
Image source, PAThe royal baby has been born on patriotic St George's Day, which celebrates the patron saint of England.
St George is heralded for his honour, bravery and gallantry. Surrounded by myth, one legend tells of St George's slaying of a dragon.
The baby prince already has an older brother George. And St George's Chapel in Windsor is where Prince Harry and Meghan Markle are marrying next month.
The royal baby shares his birthday with Lady Gabriella Windsor (pictured above with her father) - the daughter of Prince and Princess Michael of Kent, who was also born at the Lindo Wing on 23 April, 1981.
Famous names also celebrating their birthdays on Monday include actors Lee Majors, James Russo, John Hannah and Dev Patel; actress Judy Davis, singer Taio Cruz and model Gigi Hadid.
Theresa May has tweeted her congratulations to the Duke and Duchess of Cambridge.
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The BBC's Kate Whannel has been outside the Lindo Wing all morning with the dozens of royal fans waiting for the arrival of the new prince.
She said: "Champagne corks popped and cheers rang out as the news the Duchess of Cambridge had given birth to a boy ran through the crowd.
Image source, REUTERS/Henry Nicholls"Those who had been camped out for days led the chants 'it’s a boy, it’s a boy'.
"Others began singing: 'Congratulations, and celebrations. It’s a boy born on st George’s Day'.”
Historian Judith Rowbotham predicts the Duke and Duchess of Cambridge will name their new son Philip after his great-grandfather the Duke of Edinburgh.
Image source, Getty ImagesShe told the BBC: "I strongly suspect that Philip will be either a first or a second name.
"It's plain that they like the Germanic sounding names but at the same time they're looking within the Windsor traditions."
The name of the new prince is yet to be announced.
The Duke and Duchess of Cambridge are likely to wait until they have informed the Queen and the rest of their family before they tell the public the name of their third child.
Arthur, Albert and Philip are all being tipped as being in the running for the new royal baby's name.
Journalists, photographers and camera crews from around the world descended on the Lindo Wing of St Mary's Hospital in London awaiting the announcement.
Image source, EPAFans of the royal family have also been camping outside the hospital for days.
Image source, ReutersTo become King or Queen as the third-born royal child is rare - and has yet to happen within the current House of Windsor.
But the third child of George III and Queen Charlotte, William IV, took on the task and ruled from 1830 to 1837.
Image source, ReutersThe Hanoverian king acceded to the throne aged 64 when his older brother, George IV, died without an heir.
He became next in line when he was 62 and his other older brother, Frederick, Duke of York, died.
Image source, EPAThe Duchess of Cambridge has given birth to a son weighing 8lbs 7oz at 11:01 BST.
Catherine and the Duke of Cambridge's third child - who is fifth in line to the throne - was born at Lindo Wing at St Mary's Hospital, Paddington, in central London.
The arrival is the Queen's sixth great-grandchild.
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