"We could not be happier," said William, the 31-year-old Duke of Cambridge, in reference to the new prince of Cambridge, whose name is yet to be announced.
The Duchess of Cambridge, 31, gave birth to the baby boy, weighing 3.8 kilos (8 lbs 6 oz), yesterday at the private Lindo wing at St Mary's Hospital in Paddington here.
"Mother, son and father are all doing well this morning," a spokesman of the Kensington Palace, William and Kate's official residence, said in a statement.
"We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received," the Royal couple said in the statement.
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"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."
The fountain in Trafalgar Square was lit up with blue lighting to mark the birth of the third in line to Britain's throne and "It's a boy" was instantly being flashed on the BT Tower in central London.
Queen Elizabeth II was the first to be informed about her newest great-grandson by phone.
Prince Charles, the grandfather of the baby, responded to news of the birth by saying he was "enormously proud and happy" of becoming a grandparent.
"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," he said.
British Prime Minister David Cameron was among the first to take to Twitter to say: "I'm delighted for the Duke and Duchess now their son has been born. The whole country will celebrate. They'll make wonderful parents".
US President Barack Obama 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".