Celebrations are taking place across the world to mark the birth of the royal baby, delivered safely in London last night.
The Duke and Duchess of Cambridge spent the night in hospital with their new son, who was delivered at 4.24pm yesterday weighing 8lb 6oz.
The Duke was at the Duchess’s bedside when she gave birth at the Lindo Wing of St Mary’s Hospital in London.
The royal couple could leave hospital today with their son and show the baby prince to the world’s waiting media and well-wishers.
The Prince of Wales, who visited Harewood House yesterday, has become a grandfather for the first time.
He described the new arrival as an “incredibly special moment for William and Catherine”.
William telephoned his family to tell them the good news, speaking personally to the Queen, his father Charles and younger brother Prince Harry.
A Buckingham Palace spokesman said: “The Queen and Duke of Edinburgh are delighted at the news.”
The new royal baby is third in line to the throne. He will be the 43rd sovereign since William the Conqueror if, as expected, he follows reigns by Charles then William.
The Prince of Wales issued a brief statement last night saying: “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 of which was Marcus Setchell, surgeon gynaecologist to the Queen, who led the medical team that delivered Kate’s baby.
Mr Setchell spoke movingly about the baby, describing the Queen’s third great-grandchild as a “wonderful baby, beautiful baby”.
News of the birth was issued in a statement by Kensington Palace at 8.30pm yesterday: “Her Royal Highness the Duchess of Cambridge was safely delivered of a son at 4.24pm. The baby weighs 8lb 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.”
Kensington Palace issued a very brief statement from the Duke who, speaking on behalf of himself and the Duchess, said: “We could not be happier.”
A spokesman said: “The names of the baby will be announced in due course.”
A smiling David Cameron declared the birth “an important moment in the life of our nation”.
Speaking outside 10 Downing Street, the Prime Minister said last night: “It is wonderful news from St Mary’s, Paddington, and I am sure that right across the country, and indeed right across the Commonwealth, people will be celebrating and wishing the royal couple well.
“It is an important moment in the life of our nation but I suppose above all it is a wonderful moment for a warm and loving couple who have got a brand new baby boy.
“It has been a remarkable few years for our Royal Family - a royal wedding that captured people’s hearts, that extraordinary and magnificent jubilee and now this royal birth - all from a family that has given this nation so much incredible service.
“They can know that a proud nation is celebrating with a very proud and happy couple tonight.”
American president Barack Obama and his wife Michelle, who has developed a close relationship with the Queen, also spoke of their joy at the birth.
The US leader said: “Michelle and I are so pleased to congratulate the Duke and Duchess of Cambridge on the joyous occasion of the birth of their first child.
“We wish them all the happiness and blessings parenthood brings. The child enters the world at a time of promise and opportunity for our two nations.
“Given the special relationship between us, the American people are pleased to join with the people of the United Kingdom as they celebrate the birth of the young prince.”
Kate was admitted to the Lindo Wing just before 6am yesterday in the early stages of labour.
The Duchess had planned for a natural birth and had at least 10 and a half hours of labour before her baby was born. Kensington Palace said it “progressed as normal”.
Various celebrations will be held today to mark the special occasion, including gun salutes across London.
The King’s Troop Royal Horse Artillery and the Honourable Artillery Company will both carry out the ceremonial salutes in honour of the new addition to the Royal Family.
Gun salutes are fired for the birth of every prince or princess, no matter where their place is within the line of succession, the Ministry of Defence said. The last royal salute for a birth was for Princess Eugenie in 1990.
The King’s Troop Royal Horse Artillery, wearing full dress uniform, will ride past Buckingham Palace to Green Park where they will stage a 41 royal gun salute.
They will ride from their forward mounting base in Wellington Barracks into Green Park, where 71 horses pulling six First World War-era 13-pounder field guns will place them to position for the royal salute at 2pm.
Each of the six guns will fire blank artillery rounds at 10-second intervals until 41 shots have been fired. The horses and riders will then collect the guns and escort them back to Wellington Barracks.
The Honourable Artillery Company (HAC), the City of London’s Army Reserve Regiment, will fire a 62 royal gun salute from Gun Wharf at the Tower of London at 2pm.
Whilst a royal gun salute normally comprises 21 guns, this is increased to 41 if fired from a Royal Park or residence.
William and Kate’s son 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.