The couple became parents earlier this month when the Queen's granddaughter Zara gave birth to a baby girl at the Gloucestershire Royal Hospital.
She weighed 7lbs 12oz.
As with Zara's cousin Prince William and his wife Kate Middleton when they welcomed son George, the baby's name was not immediately released to the public.
Mia will follow in Zara's footsteps and not have a royal title, instead being known as Miss Tindall.
She is sixteenth in line to the throne.
A statement following the birth said The Queen, the Duke of Edinburgh, Zara's parents and Mike's parents were "thrilled" by the arrival, while former rugby player Mike tweeted: "Thanks so much for all your kind messages. Definitely the best day of my life today, so happy.! The girls are both doing great!"
Zara and Mike announced their pregnancy in July 2013. They met in 2003 during England's Rugby World Cup campaign and married eight years later in Canongate Kirk, Edinburgh.
Celebrity babies who have arrived in 2014:
Copyright: Rex Features WENN/Matthew Taylor/Daniel Deme/Josiah True/Matthre Taylor/Joe