The late Diana moved into Kensington Palace after marrying Prince Charles in 1981. William and Harry were raised there.
When she died in 1997, mourners laid flowers and tributes outside the palace gates.
On Tuesday, London reverberated with the sound of cannon fire and peals of bells to mark the birth.
Shortly before the new baby's departure from St. Mary's, Prince Charles stopped by for a brief visit with his first grandchild, accompanied by his wife, Camilla, the Duchess of Cornwall. He told reporters it was "marvelous."
And Catherine's parents, Carole and Michael Middleton, visited earlier. As grandmothers tend to do, she described her grandbaby as "absolutely beautiful."
She and her husband were "so thrilled" about being grandparents.
"It was so exciting. It was fantastic," said Eliza Wells, one of the well-wishers gathered outside the hospital. "The crowd erupted, because everyone's been waiting so long for it."
William and Catherine "both seemed very relaxed, even with the press there and the crowd," Wells said. "They just seemed like a normal couple."
A normal life?
Royal commentators said the Duke and Duchess of Cambridge will try to give their son as regular an upbringing as possible.
But the intense media interest in the birth of the new prince highlights the challenge his parents face in trying to protect his privacy and maintain a degree of normalcy.
"This baby has two things stopping it from being normal," historian Kate Williams said. "No. 1, it lives in a life of incredible wealth and privilege. ... No. 2, it is an incredible celebrity, and we've seen this with the coverage."
But Prince William loved that his mother tried to give him as normal a childhood as possible, including trips to the cinema and an amusement park, and sending him to a local private school as a boy. "And that's what he wants for little baby Cambridge," Williams offered.
Although the excitement over his birth is not universal, there's no doubting the level of global interest in the prince.
On Monday, there were more than 19 million Facebook interactions related to the royal baby, according to the site.
His birth also took Twitter by storm.