The newborn was delivered at 05:26 BST on Monday, with Prince Harry saying the baby boy was "absolutely to die for".
The Palace said the Queen, the Duke of Edinburgh and the Prince of Wales were all "delighted with the news".
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 they planned to present the baby to the media in two days' time "so everyone can see" him.
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, while Alexander and James are also popular.
The baby will not automatically become a prince - although he could get that title if the Queen steps in.
The baby weighed 7lbs 3oz (3.2kg), making him lighter than each of his cousins.
Prince George was 8lb 6oz (3.8kg) when he was born in July 2013. His sister Princess Charlotte was 8lbs 3oz (3.7kg) when she arrived in May 2015, while their younger brother Prince Louis weighed 8lb 7oz (3.8kg) when he was born in April last year.
The average weight of a baby in the UK is about 7lb 7oz (3.4kg).
A statement 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.
Kensington Palace said the Duke and Duchess of Cambridge were "delighted" and "looking forward" to meeting him.
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.
The choir that sang at Prince Harry and Meghan's wedding wrote a lullaby for the new baby.
Media captionThe Kingdom Choir sang Stand by Me at Prince Harry and Meghan's wedding in 2018
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.
Meghan's mother is African-American and her father is Dutch-Irish, so the infant is believed to be the first mixed-race child born to a senior member of the British Royal Family in centuries.
Celebrating mixed-race identity
Royal commentator Joe Little said the baby's "cocktail" of heritage makes the royals "more relevant".
"Many people all over the UK have a mixed heritage so this reflects life in the 21st Century," he said.
"Some would argue that this is a long time coming but it just shows that the Royal Family nowadays marries for love, and this is a natural progression," he added.
"It will make the British Royal Family more relevant, which can only be a good thing."
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."
Meanwhile New Zealand Prime Minister Jacinda Ardern said a donation would be made to a group which prepares baby boxes for families in need, and 10 native trees would be planted in New Zealand in honour of the royal baby.
Australian Prime Minister Scott Morrison also sent his "huge congratulations" to the couple.
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.
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