The Sussex baby was born at 5.26am this morning and weighs 7lbs 3oz.
The baby's sex was a surprise for the delighted couple, who chose not to find out what they were having.
The newest addition to the Windsor family is a first cousin of the Duke and Duchess of Cambridge's children, Prince George, Princess Charlotte and Prince Louis, and a fourth grandchild for the Prince of Wales.
Baby Sussex's arrival comes less than a year after Harry married American former Suits actress Meghan in a glittering ceremony in St George's Chapel, Windsor Castle.
Speaking from Windsor, the Duke of Sussex said: "I'm very excited to announce that Meghan and myself had a baby boy this morning, a very healthy baby boy."
He said: "It's been the most amazing experience. We are both absolutely thrilled and grateful to the support from everybody out there."
Prince Harry said they were still thinking about names but would show off their baby to the world in the two days' time.
The Duke of Sussex added: "It's absolutely incredible and I'm so incredibly proud of my wife."
Harry, who was present for the birth, said his son was "absolutely to die for...I'm absolutely over the moon".
The royal couple are currently at their new home, Frogmore Cottage, close to the Queen's Windsor Castle home.
The Duchess’s mother, Doria Ragland, who is overjoyed by the arrival of her first grandchild, is with the royal couple at Frogmore Cottage.
Sources close to the family said Ms Ragland wanted to be by her daughter's side for the event.
Ahead of the birth, Harry and Meghan made a personal decision to keep their arrangements private, amid speculation they had opted for a home birth at Frogmore Cottage.
They also said they would only share the news of the baby's arrival once they had had a chance to celebrate privately as a family.
Reports had previously suggested Meghan could opt for the Kensington wing of the Chelsea and Westminster Hospital where pals George Clooney's wife Amal had their children or an NHS hospital in Surrey, the Frimley Park hospital which is just 15 miles away from their home.
The Sun on Sunday revealed April 28 was Meghan's official due date - but Prince Harry, 34, surprised fans when he arrived at the London Marathon last week to hand out medals.
But Harry was forced to cancel part of a planned trip to the Netherlands this week amid rumours Meghan's first baby was due last weekend. The duke has an official visit to The Hague scheduled for Thursday when he is due to launch the one-year countdown to the Invictus Games 2020.
00:00 I don't know myself had a baby boy early
00:03 this morning
00:03 a very healthy boy mother and baby are
00:07 doing incredibly well it's been the most
00:10 amazing experience I can ever possibly
00:14 imagine how any woman does what they do
00:18 is beyond comprehension but we're both
00:22 absolutely thrilled and so grateful to
00:25 all the love and support for everybody
00:27 out there from everybody out there it's
00:29 been it's been amazing so we just wanted
00:31 to share this with everybody still think
00:35 about names on it's the babies a little
00:38 bit overdue so we've had a little bit
00:40 time to think about it but some yeah
00:42 we're still that's that's but that's the
00:44 next fit but for us I think we'll be
00:46 seeing you guys in probably two days
00:48 time as planned
00:49 as a family to be able to share it with
00:51 you guys and so one can see the baby I
00:59 haven't been in many births this is
01:03 definitely my first birth it was amazing
01:06 absolutely incredible and as I said I'm
01:07 so incredibly proud of my wife and as
01:11 every father and parent would ever say
01:14 you know you're your babies absolutely
01:16 amazing but this little thing is
01:18 absolutely to die for so I'm just over
01:20 the moon thank you very much guys
01:30 well this
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