Talking about their first few days as parents, Meghan said: "It's magic, it's pretty unbelievable".
The Duke and Duchess of Sussex showed their newborn to the world at Windsor Castle just after noon local time. In fact, the speculated top pick among British oddsmakers was Arthur, followed closely by James, Philip and Albert.
The new royal will be meeting his great-grandparents Queen Elizabeth and the Duke of Edinburgh later this afternoon in Windsor.
The Duchess said it had "been a special couple of days" and thanked well-wishers for the outpouring of support they had received.
Throughout their pregnancy, several reports - and some alleged royal sources - claimed the Duke and Duchess of Sussex want a more normal life for their baby Prince Harry has had a life-long love-hate relationship with the "fishbowl" of Kensington Palace and royal life and, understandably, wants something different for his children.
The baby boy, who is seventh in line to the throne, will be called Archie Harrison Mountbatten-Windsor.
Prince Harry and Meghan Markle hold their son, Archie Harrison.
Harry continued: "It's great".
In a significant milestone both across the Commonwealth and within British society, the newborn has made history by becoming the first biracial British child in the royal family.
This decision, coupled with the launch of the Duke and Duchess's official Instagram account days ahead of Meghan's due date, suggested they may choose to share the first picture of him there.
"I don't know who he gets that from", joke Harry.
Baby Sussex's birth comes less than a year after Harry married American former actress Meghan in a glittering ceremony in St George's Chapel, Windsor Castle. The 93-year-old monarch also has eight grandchildren, and four children.
The announcement came after the Queen and Duke of Edinburgh met their eighth and newest great-grandchild for the first time at Windsor Castle.
"Obviously thrilled, absolutely thrilled, and obviously looking forward to seeing them in the next few days", William told reporters.