The Duchess of Sussex has given birth to a boy, the Duke of Sussex has announced.
A beaming Prince Harry said they were "absolutely thrilled" and thanked the public for their support.
He said Meghan and the baby were doing "incredibly well", adding that they were still thinking about names for the infant, who was delivered at 05:26 BST.
Buckingham Palace said the baby weighed 7lbs 3oz (3.2kg), and that the duke was present for the birth.
The baby boy is seventh in line to the throne, behind the Prince of Wales, the Duke of Cambridge and his children - Prince George, Princess Charlotte and Prince Louis - and Prince Harry.
He is the Queen's eighth great-grandchild.
Speaking after the birth, Harry said the child had been "a little overdue" and that he planned to make another announcement in two days' time "so everyone can see the baby".
Asked what it was like to be present for the birth, he laughed and said: "I haven't been at many births. This is definitely my first birth. It was amazing, absolutely incredible, and, as I said, I'm so incredibly proud of my wife.
"As every father and parent will ever say, you know, your baby is absolutely amazing, but this little thing is absolutely to-die-for, so I'm just over the moon."
He added: "It's been the most amazing experience I could ever have possibly imagined.
"How any woman does what they do is beyond comprehension."
A framed notice of birth has been placed on display on a ceremonial easel on the forecourt of Buckingham Palace, where it will remain until 20:00 on Monday.
Buckingham Palace added: "The Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall, the Duke and Duchess of Cambridge, Lady Jane Fellowes, Lady Sarah McCorquodale and Earl Spencer have been informed and are delighted with the news.
"The duchess's mother, Doria Ragland, who is overjoyed by the arrival of her first grandchild, is with their Royal Highnesses at Frogmore Cottage."
In an Instagram post, Kensington Palace said the Duke and Duchess of Cambridge were "delighted" at the news and "look forward to meeting the latest addition to the family".
The baby is a dual citizen, having automatically acquired US citizenship, via Meghan, at birth.
Speaking to the Sun, Meghan's father, Thomas Markle, congratulated the new parents and spoke of his pride at the birth.
"I am proud that my new grandson is born into the British royal family and I am sure that he will grow up to serve the crown and the people of Britain with grace, dignity, and honour," he said from his home in Mexico.
Charles Spencer - brother of Diana, Princess of Wales and uncle to Harry - tweeted his congratulations.
He wrote: "Really very lovely news today - many, many, congratulations!"
The fine line between public and private life
By BBC royal correspondent Jonny Dymond
So what's going on with the birth of the Duke and Duchess of Sussex's first child?
Buckingham Palace announced some weeks ago that there would be no information given out about the birth, beyond that it was happening.
And so it was that shortly before 14:00 BST on Monday, a brief statement from Buckingham Palace announced that Meghan had gone in to labour, followed 40 minutes later by confirmation of the baby's arrival - a boy, weighing 7lbs 3oz.
That meant the strange British circus of journalists, photographers, royal superfans and bemused passers-by gawking at a hospital door for days on end would not happen.
Responding to the news, Prime Minister Theresa May tweeted: "Congratulations to the Duke and Duchess of Sussex on the arrival of their baby boy. Wishing you all the best at this happy time."
Labour leader Jeremy Corbyn added: "Congratulations to Meghan and Harry on the birth of their baby. I hope they're all doing well."
And Scotland's First Minister Nicola Sturgeon also offered her congratulations to the couple.
She wrote: "The birth of a baby is a joyful occasion - congratulations to Harry and Meghan."
Archbishop of Canterbury Justin Welby, who married the couple in May last year, said: "May God bless the new family with love, health and happiness."
Harry and Meghan, a timeline
8 November 2016 - Kensington Palace releases a statement that confirms Prince Harry has been dating Meghan Markle "for a few months" and asks the press to respect their privacy
28 November 2017 - Harry and Meghan announce they are engaged to be married
15 December 2017 - Kensington Palace confirms the couple have chosen to wed in Windsor on 19 May the following year
19 May 2018 - Harry and Meghan are married in front of 600 guests at St George's Chapel and become the Duke and Duchess of Sussex
15 October 2018 - Kensington Palace announces the duchess is pregnant, and is due to give birth in Spring 2019
6 May 2019 - Meghan gives birth to a boy, who becomes seventh in line to the throne