Meghan Markle went into labour on Monday morning and has given birth to a boy.
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The duke and duchess's Instagram account @SussexRoyal also announced the birth saying: "It's a boy! Their Royal Highnesses the Duke and Duchess of Sussex are overjoyed to announce the birth of their child."
A beaming Harry said: "It's been the most amazing experience I could ever have possibly imagined.
"How any woman does what they do is beyond comprehension. But we're both absolutely thrilled.
"We're both absolutely thrilled and so grateful for all the love and support from everybody out there, it's been amazing, so we just wanted to share this with everybody."
Harry said the couple were "still thinking about names" for their son.
He said: "The baby was a little bit overdue so we've had a little bit of time to think about it.
"That's the next bit."
He said he planned to make another announcement in two days' time "so everyone can see the baby".
Harry added: "I haven't been at many births.
"This is definitely my first birth - it was amazing, absolutely incredible.
"I'm so incredibly proud of my wife.
"And as every father and parent would say, your baby is absolutely amazing.
"But this little thing is absolutely to-die-for so I'm absolutely over the moon."
Harry and Meghan's baby will be seventh in line to the throne and the Queen and the Duke of Edinburgh's eighth great-grandchild, and a fourth grandchild for the Prince of Wales.
The baby's sex will be a surprise for the couple, who chose not to find out what they were having.
Ahead of the birth, the duke and duchess made a personal decision to keep the arrangements private, amid speculation they had opted for a home birth at their new home Frogmore Cottage, close to the Queen's Windsor Castle home.
They have said they will only share the news of the baby's arrival once they have had a chance to celebrate privately as a family.
Meghan was last seen on an official engagement on March 19 when she signed a book of condolences in London with Harry for the victims of the Christchurch terror attack.
The American former actress and the duke moved into their renovated home Frogmore Cottage on the Windsor Estate at the beginning of April, as they prepared for their baby's arrival.