Pippa and her husband James attended the wedding of Princess Beatrice and Jack Brooksbank on Friday.
No announcement has been made yet about the name of Pippa's baby, but the bookies' favourite is Michael.
St. Mary's is the same hospital where her older sister, Catherine, Duchess of Cambridge, delivered her children.
A source close to the family said: 'Everyone is delighted and mother and baby are doing well'. Despite being nine months pregnant, she slipped into a chic pair of heels, topping off the look with a wide-brimmed green hat.
Middleton and Matthews have visited the Paddington-based hospital several times over the past few months, with the expectant mum's last visit being just last week.
Spencer has always declined to speak about his brother and Pippa's pregnancy in public, pointing out they are a very private couple.
The news was announced on Royal correspondent Rebecca English's Twitter account, as she wrote: "George, Charlotte and Louis have a new cousin!"
"The Duke and Duchess of Cambridge are thrilled for Pippa and James", the Palace said.