They also set up their own court at Buckingham Palace in another move to separate themselves from Prince William and Kate Middleton's household.
Archie's maternal grandmother, Meghan's mom Doria Ragland, has been staying with the couple during their first days as parents.
She said: "He said he's getting used to the baby and how Archie has fitted into family life".
The mother-of-three, from Toronto, is close friends with Meghan, 37, and is believed to be a front runner to become the royal baby's godmother.
Having Doria for support through any first-time parenting nerves has been a big help for Meghan. "They are so incredibly happy".
New dad Prince Harry has also been busy, jetting off to the Netherlands on May 9 - just three days after his son's birth - to officially launch the countdown to the 2020 Invictus Games. The couple are understood to have visited Harry and Meghan and their newborn son at Frogmore Cottage in Windsor this afternoon.
"We honor and celebrate each and every one of you", the post added. Today is Mother's Day in the United States, Canada, Australia, New Zealand, South Africa, Kenya, Japan, and several countries across Europe.
The Duke of Sussex turned up today for his second full day of engagements since the arrival of the seventh in line to the throne, confessing that the presence of the smallest member of the royal family in the house was not necessarily compatible with the world of work.
Speaking to the Mirror, retired rugby star Mike described the meeting at Highgrove as "carnage", saying: "I don't think Mia will remember that day, she was two months old, but it was great, it was carnage".