Zara, the Queen’s grand-daughter, gave birth at Stroud Maternity Unit in Gloucestershire on Monday.
The new baby, whose name is yet to be confirmed, will be 19th in line to the throne and the Queen’s seventh great-grandchild.
Buckingham Palace said in a statement that Mr Tindall was present at the birth.
They added: “The Queen, the Duke of Edinburgh, the Princess Royal, Captain Mark Phillips and Mike’s parents, Mr Philip and Mrs Linda Tindall, have been informed and are delighted with the news.”
The little girl is a younger sister to four-year-old Mia Tindall and a second cousin to Prince George, Princess Charlotte and Prince Louis.
The new arrival is not an HRH and will not have a title.
The birth comes just over a year since Zara, a champion equestrian rider who won a team silver medal in the London 2012 Olympics, suffered a miscarriage.