Zara Philips and Mike Tindall have spent their first night at home with the new baby girl.
After Zara gave birth yesterday morning, the family left hospital during the afternoon, and are thought to have headed home on the Princess Royal's Gatcombe Park estate near Stroud, Gloucestershire.
Tindall took to Twitter to describe his joy at becoming a father for the first time, saying it was "definitely the best day of my life", adding that mother and baby were doing well.
Zara, a silver medal-winning Olympian, who is one of Britain's leading equestrian riders, gave birth to her first child at Gloucestershire Royal Hospital with her husband, the former England rugby star, at her side.
The new dad 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!"
A few minutes later he praised the treatment his wife had received at hospital: "Big thanks to all involved at Gloucester Royal, they were fantastic!"
The couple left hospital with their royal baby who weighed 7lbs 12oz, is 16th in line to the throne and is the Queen's fourth great grandchild.
Buckingham Palace said in a statement: "The Queen, the Duke of Edinburgh, the Princess Royal, Captain Mark Phillips and Mike's parents, Mr Phillip and Mrs Linda Tindall, have been informed and are delighted with the news.
"The baby's name will be confirmed in due course."
Prime Minister David Cameron sent a message to the happy couple in a tweet - using the hashtag #royalbaby - to say: "Many congratulations to Zara and Mike Tindall on the birth of their baby girl."
The Princess Royal carried on as normal at Gatcombe Park, following the birth of her third grandchild.
Anne, who was wearing a long coat, hat and dark glasses, drove up to a gatehouse near the entrance to the estate and collected several dogs before putting them into the 4x4 and driving back down the lane away from the main road.
Zara's pregnancy was confirmed by the Palace in July, just a few weeks before the birth of the Duke and Duchess of Cambridge's baby son Prince George.
She took a tumble while competing at the Barbury International Eventing and Show in Marlborough, Wiltshire, a few days before her pregnancy became public knowledge.
But she was described as ''fine'' by a spokeswoman for Juniper Sport, a PR and sports agency that represents her.
Zara won a silver medal at London 2012 with eventing equestrian teammates William Fox-Pitt, Mary King, Tina Cook and Nicola Wilson, and is planning to take part in the Rio de Janeiro Games in 2016.
She married former rugby international Tindall in July 2011 in a ceremony staged in Edinburgh and attended by the Queen and senior royals.
The baby girl follows Zara's brother Peter Phillips' two children and Prince George to become the Queen and the Duke of Edinburgh's fourth great grandchild.
Peter Phillips' daughters Savannah and Isla Phillips were also born at Gloucestershire Royal Hospital in Gloucester.
Zara and Tindall moved back to Anne's Gatcombe Park estate last January after selling their regency home in Cheltenham.
The pair first met in a Sydney bar when Tindall was in Australia for the 2003 Rugby World Cup.
In the days leading up to the birth of their baby daughter, Zara - who is also a horse trainer - was spotted watching three of her entrants run in the Heythrop Hunt Point-to-Point races in Aldsworth, Gloucestershire on January 12.
Bookmakers William Hill are offering odds of 3/1 that Zara and Tindall will name their daughter Charlotte, 5/1 that she will be called Victoria, 6/1 Anne after her grandmother and 7/1 Elizabeth after her great-grandmother.
But the public has mostly backed the name Emma which has odds of 10/1.
Punters can also get odds of 50/1 that the baby will one day represent Team GB at the Olympics or 33/1 that she will play international rugby.