Marcus Setchell, who delivered the Duke and Duchess's son, said the tot was "a wonderful baby, a beautiful baby".
The Duke of Cambridge looks likely to come out of the Lindo Wing holding his newborn son (4/7), or it's a 6/4 chance that Kate does.
The future King is favoured with the firm at 8/11 to ascend the throne before 2065, and it's also odds-on that the couple announce their son is expecting a brother or sister (8/11) before Prince Harry gets married (2/1).
Jessica Bridge of Ladbrokes said: "James is the red-hot favourite and the odds suggest we'll get the name on Tuesday. We're staring down the barrel of a gun if the baby's named Henry."
She added: "It's been odds-on all year long that Kate would give birth to a girl and punters who believed it would be a boy have now collected over £100,000 worth of winnings."
Now the little Prince is finally here, the odds are flooding in on what he will be called.
Kate and William are favoured to name their baby boy James, according to Ladbrokes.
The name, shared by Kate's younger brother, is the red-hot 2/1 favourite.
George is fancied as a 5/1 shot, and it's the same price for the name Henry, after an unprecedented amount of bets in the last 48 hours saw Henry's odds slashed from 50/1 to 5/1.
It's odds-on that the name is announced on Tuesday 23rd July, but for those who think we could be waiting a week like William's announcement can take odds of 3/1 on offer.
The statement, on Buckingham Palace headed paper, was brought out for public inspection on the easel by Badar Azim, a footman with the Royal Household, and Ailsa Anderson, the Queen's press secretary.
The easel, in the Rococo revival style, was previously used to announce the birth of Prince William in 1982.
It is a carved gilt wood easel with moulded uprights ending at scrolled cabriole legs and feet.
Earlier, the Prime Minister said the "whole country will celebrate" the birth of the royal baby as news of the child's arrival spread across the globe.
David Cameron offered his congratulations on Twitter where the worlds of showbiz, politics and sport appeared to share in the Duke and Duchess's delight.
He wrote: "I'm delighted for the Duke and Duchess now their son has been born.
"The whole country will celebrate. They'll make wonderful parents."
The Prince of Wales issued a brief statement: "Both my wife and I are overjoyed at the arrival of my first grandchild. It is an incredibly special moment for William and Catherine and we are so thrilled for them on the birth of their baby boy.
"Grandparenthood is a unique moment in anyone's life, as countless kind people have told me in recent months, so I am enormously proud and happy to be a grandfather for the first time and we are eagerly looking forward to seeing the baby in the near future."
Moments before Charles's joyous words were released the Queen's press secretary Ailsa Anderson strode out onto Buckingham Palace's forecourt to place the official notice on its ornate stand.
The document had four signatures, first was that of Marcus Setchell, Surgeon Gynaecologist to Queen, who led the medical team that delivered Kate's baby.
Other members of his team included Guy Thorpe-Beeston, obstetrician and Dr Sunit Godambe, consultant neonatologist at St Mary's Hospital.
Kensington Palace said "The names of the baby will be announced in due course."
Prime Minister David Cameron has passed on his well wishes to the Duke and Duchess of Cambridge.
He spoke of the "wonderful moment for a young couple and their baby boy".
Confirmation of the royal birth arrived by car in the form of an official statement, carried by the Duke and Duchess of Cambridge's press secretary Ed Perkins.
It was met by a footman before being taken inside the palace.
Another visitor to the Palace, 26-year-old German Veronika Schwarz, said she thought there was greater excitement outside of England.
"We don't have anything like the Royal Family in Germany, so we are all getting quite carried away with this."
A handful of royal fans have spent days camped outside the Lindo Wing of St Mary's hospital, eager to snatch a glimpse of the baby and future sovereign.
Glaswegian couple Douglas Johnstone and Victoria Wallace said they felt fortunate that their trip to London coincided with the royal birth.
Mr Johnstone said: "We are very lucky to have planned our trip at this time."
Police chatted with visitors, many pondering potential names for the newborn future monarch.
Matthew and Donna Harold, from Michigan in the United States, said they had been asked to stock up on commemorative souvenirs for friends back home during their holiday in London.
"The royal baby is front-page news every day back in the States at the moment, there is a lot of excitement," said Mrs Harold.
"Our holiday was booked months ago, so we did not expect to be over here while the baby was happening, so we have been told to bring back as many newspapers and souvenirs with the baby on as possible."
In Australia, a set of commemorative baby stamps has already been commissioned to mark the royal birth.
Lynette Traynor, a postal worker from Melbourne who is on holiday in London, said: "We love everything to do with the royals, so the news is full of it.
"We have a set of baby stamps ready to go, as soon as it has been confirmed. We can't get enough of it."
Well-wishers from around the globe screamed with excitement as news of the birth of the royal baby spread to Buckingham Palace.
One man shouted: "It's a boy," prompting crowds to erupt into spontaneous cheering, ahead of the easel being placed in the forecourt.
Three were three cheers of "hip hip hooray," as the crowd outside the palace gates swelled to more than 10 deep.
Many had been waiting patiently for confirmation of the birth ever since news of the Duchess going into labour this morning.
The months of speculation and anticipation - dubbed the Great Kate Wait - built to a climax this morning amid news that the Duchess of Cambridge had been admitted to hospital in the early stages of labour.
The Prince of Wales said that he was "enormously proud and happy to be a grandfather for the first time", adding that it was "an incredibly special moment for William and Catherine".
Kate has given birth to a baby boy - fulfilling her wish for a son.
When she was mid-way through her pregnancy, the Duchess told a soldier at a St Patrick's Day parade in Aldershot, Hampshire, that she did not know the sex of her baby, but that she would like a boy and Prince William a girl.
Guardsman Lee Wheeler, 29, said: "I was talking to her about the baby, of course. "I asked her 'Do you know if it's a girl or boy?', and she said 'Not yet'.
"She said 'I'd like to have a boy and William would like a girl'. That's always the way."
The new royal baby is the first Prince of Cambridge to be born for more than 190 years.
The last was Prince George of Cambridge, a grandson of George III and the only son of Prince Adolphus Frederick, the 1st Duke of Cambridge.
Prince George of Cambridge was born in 1819 and refused to have an arranged marriage. He wed a commoner for love after falling for the actress Sarah Louisa Fairbrother, who was said to be a classic beauty and a graceful dancer.
They married in 1847 when she was already the mother of two of his children and pregnant with his third.
But the Duke did not seek the sovereign's approval and the marriage was never recognised, hence his children were not eligible to inherit royal titles.
Miss Fairbrother was ostracised by the Royal Family and never given a title. Instead she became known by the nickname Mrs FitzGeorge and this surname was taken by George's offspring.
Despite his marriage, George had a wandering eye and soon after he wed he took up with mistress Louisa Beauclerk, who remained his lover for more than 30 years.
He went on to become the 2nd Duke of Cambridge after his father's death.
The Duke was in the Army and served in the Crimean War. He was promoted to Commander-in-Chief in 1887 and an equestrian statue of him stands in the middle of London's Whitehall.
He was said to have been a disciplinarian, who believed Army promotions should be based on social connections rather than ability. He died in 1904.
His father, the 1st Duke of Cambridge - who lived from 1774 to 1850 - was never a Prince of Cambridge, but was given the title the Duke of Cambridge by his father George III in 1801 when he was 27.
The notice has been placed on the easel outside Buckingham Palace giving formal notification of the royal baby's birth.
The message read: "Her Royal Highness the Duchess of Cambridge was safely delivered of a son at 4.24pm.
"HRH and her child are both doing well."
The Prince of Cambridge will be the Queen's third great-grandchild and is destined to be crowned monarch.
He will be the 43rd sovereign since William the Conqueror if, as expected, it follows reigns by the Prince of Wales then William.
It is understood that Prince William telephoned the Queen from the Lindo Wing to inform her of his son's birth.
He then spoke to Prince Charles and Prince Harry.
The Middletons were also informed before the information was released to the media.
Soon after the palace announcement was made, a notice giving details about the baby - the third-in-line to the throne - left St Mary's Hospital in London by car for Buckingham Palace.
It has been signed by the Queen's former gynaecologist Marcus Setchell, who led the medical team that delivered Kate's baby.
The notice will be placed on an easel in the palace's forecourt just like the announcement of William's birth on June 21, 1982, a traditional element of theatre in marked contrast to the modern age of emails and Twitter.
Buckingham Palace said in a statement: "Her Royal Highness The Duchess of Cambridge was safely delivered of a son at 4.24pm. The baby weighs 8lbs 6oz.
"The Duke of Cambridge was present for the birth.
"The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news.
"Her Royal Highness and her child are both doing well and will remain in hospital overnight."
A Buckingham Palace spokesman said: "The Queen and Duke of Edinburgh are delighted at the news."
The announcement of the birth has now been handed over into a car to be taken to Buckingham Palace.
It will be displayed on an easel shortly.
And the crowds are already shouting three cheers.