The cottage had been a royal gift to Abdul Karim by Queen Victoria, then Empress of India, in recognition of his service and as a sign of her affection towards her Indian aide and confidant.
Prince Harry, Duke of Sussex, was present at the birth of his boy, who weighs 3.2 kg.
During a four-hour-long heart to heart over lunch at Windsor Castle on Sunday, the 93-year-old monarch told the 35 -year-old Duke of Sussex that he was a "much loved" member of her family and that he will be welcome back with open arms if he changes his mind in future.
There's so much left unsaid in the Netflix documentary, Harry & Meghan.
'It's only been two and a half days, three days, but we're just so thrilled to have our own little bundle of joy'
Harry, 35, was said to be desperate to be reunited with Archie, whom he had not seen for two weeks, and his 38-year-old American actress wife in Canada that the young family plans to make their home.
"The Sussexes will not use their HRH titles as they are no longer working members of the Royal Family. The Duke and Duchess of Sussex have shared their wish to repay Sovereign Grant expenditure for the refurbishment of Frogmore Cottage (believed to be 2.4 million), which will remain their United Kingdom family home," a statement from the Buckingham Palace read.
The Duke and Duchess of Sussex are planning to work towards becoming 'financially independent'.
His remarks come a day after Buckingham Palace announced Harry and Meghan, who had announced their decision to step back from their royal duties, will not use the "royal highness" titles as well as would no longer receive public funds for royal duties.