Congratulations have poured in from around the world following the birth of the Duke and Duchess of Sussex's son.
The newborn was delivered at 05:26 BST on Monday, with Prince Harry saying the baby boy was "absolutely to-die-for".
Kensington Palace said the Duke and Duchess of Cambridge were "delighted" and "looking forward" to meeting him.
Meghan's mother, Doria Ragland, was said to be "overjoyed", while former US First Lady Michelle Obama said she and Barack were "thrilled".
The baby, who has not been named yet, 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 he planned to make an announcement in two days' time "so everyone can see the baby".
He said Meghan and the baby were doing "incredibly well", adding that they were still thinking about names for the infant.
Arthur - a traditional royal name that is among the middle names of the baby's grandfather, the Prince of Wales, and his uncle the Duke of Cambridge - is the favourite with many of the bookies.
Buckingham Palace said the baby weighed 7lbs 3oz (3.2kg), and that the duke was present for the birth.
In a statement the Palace said that the Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall and the Duke and Duchess of Cambridge were all "delighted" with the news.
It added that Meghan's mother was "overjoyed at the arrival of her first grandchild" and was with her daughter at Frogmore Cottage - the Sussexes' home on the Windsor Estate.
Prime minister Theresa May, Labour leader Jeremy Corbyn and Scotland's First Minister Nicola Sturgeon were among the politicians to offer their congratulations to the royal couple on Twitter.
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."
Congratulations also came from celebrities and famous figures in the United States, where Meghan was born.
Patrick J Adams, who starred alongside Meghan in the US legal drama Suits, wrote on Twitter: "Just heard that the world just got heavier by 7 pounds and 3 ounces. Much love to him and his incredible parents."
The baby is a dual citizen, having automatically acquired US citizenship, via Meghan, at birth.
Speaking to the Sun from his home in Mexico, Meghan's father, Thomas Markle, said he was "proud" his grandson was born into the British royal family and he was "sure that he would grow up to serve the crown and the people of Britain with grace, dignity, and honour."
The fine line between public and private life
By BBC royal correspondent Jonny Dymond
The Duke and Duchess of Sussex will have decisions to make about just how royal they want their child to be - in title, upbringing and public exposure.
And the couple have a difficult line to tread between their public life and the life they would prefer to remain unseen.
That line, between the royals' public and private life, has shifted over the decades.
In his announcement on Monday, a beaming Harry said he was "over the moon" and he was "incredibly proud" of his wife, adding: "How any woman does what they do is beyond comprehension."
The duke is due to travel to the Netherlands on Thursday for the launch of the 2020 Invictus Games in The Hague.
Meghan and Harry have asked the public to donate to four children's charities - Little Village, the Lunchbox Fund, Well Child, and Baby2Baby - instead of sending presents for the baby.
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