The boy, weighing 8lbs 60z, was born at 4.24pm, with Prince William present.
He will be third in line to the throne after Prince Charles and Prince William.
The baby, which will eventually be King, will be styled HRH Prince (forename) of Cambridge and, in Scotland, the Master of Strathearn.
Should he require a surname, such as on marriage, it will be Mountbatten-Windsor.
Kensington 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."
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 was 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.
The new royal baby 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.
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 on to 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."
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...the whole Commonwealth... will celebrate. They'll make wonderful parents."
The Duke and Duchess of Cambridge are expected to pick a traditional royal name for their newborn baby Prince.
Royal infants mostly have historical names which are passed down through the generations.
Charles Kidd, editor of Debrett's Peerage and Baronetage, predicted that William and Kate would select a conventional first name for their son.
"I do think that Prince William is quite a traditionalist - that's my feeling," Mr Kidd said.
"The fact that he chose to use the royal dukedom (of Cambridge) and has followed royal precedent, I wouldn't be surprised if they use a name that is already familiar in the royal family - something that has royal associations."
He suggested that George - a favourite at the bookmakers - could be their name of choice. There are also predictions that the couple will honour William's father the Prince of Wales by choosing Charles and the Duke of Edinburgh by selecting Philip.
Midway through her pregnancy, Kate revealed she and William had a shortlist of names and that her friends were giving her ideas.
"We have a shortlist for both (boy and girl) but it's very difficult. My friends keep texting me names," she told a well-wisher.
One day the baby is also likely to become the Prince of Wales. The title is usually given to the male heir to the throne, but it is not automatic.
Royal babies also usually have a number of middle names. The Prince of Wales has four names, as does William.