It's unclear whether the birth of Meghan and Harry's child will be announced via an easel outside Buckingham Palace, as was the case with the Duke and Duchess of Cambridge's three children.
Labour leader Jeremy Corbyn said: "Congratulations to Meghan and Harry on the birth of their baby".
They were swiftly followed by the Archbishop of Canterbury, who officiated at their wedding nearly a year ago.
The post refers to Queen Elizabeth II, her husband Prince Philip, their son and Harry's father Prince Charles, his wife Camilla, Harry's brother Prince William and wife Kate Middleton, and Diana's two sisters and brother-Baby Sussex's great-aunts and great-uncle.More news: PM Narendra Modi praises Naveen Patnaik for cyclone handling
This is definitely my first birth. "As I say, I am so incredibly proud of my wife and, as every father and parent would ever say your baby is absolutely incredible, this little baby is absolutely to die for".
The new dad said that he and Markle are still thinking about what to name their son. "The Duchess and baby are both healthy and well, and the couple thank members of the public for their shared excitement and support during this very special time in their lives", the caption said.
Prince Harry said the couple plans to appear publicly with their son in the coming days.
The fact that details of the birth were kept so private is thought to reflect Harry's hard relationship with the media over the years and his wish to protect the privacy of his wife and child as much as possible.
Clive Spence-Jones of Whittington Hospital in North London told The Daily Mail: "A recent study found that women over 36 should consider induction around their due date, as they are at higher risk. It's been wonderful, so we just wanted to share this with everybody".More news: Markets plunge after Trump threatens new China tariff hike
The baby was safely delivered at 05:26 but it has not been confirmed as to whether Meghan had him at home at Frogmore Cottage in Windsor or in hospital. "The baby weighs 7-pounds, 3-ounces".
The child's rank could fall further if the Duke and Duchess of Cambridge have more children.
Meanwhile 5,500 miles away at Frogmore Cottage in Windsor, Meghan's mother Doria was by their daughter's side - and even got a mention in the official Palace statement.
The baby has now, with immediate effect, taken over Prince Andrew's role and become seventh in line to the throne.
It's understood that the child will not be a prince or princess, unless the Queen issues a new Letters Patent.More news: SpaceX launches supplies to space station after power delays
Had the baby been a girl, it could have been Lady (first name) Mountbatten-Windsor.