Mike Tindall has described his joy at becoming a father for the first time, tweeting that the birth of his daughter was "definitely the best day of my life".
His wife Zara Phillips gave birth early this morning and Tindall used the micro blogging site to report 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 earlier 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."
Tina Cook, who won an eventing team silver medal with Zara at the London Olympics, said: "I'm absolutely thrilled for her, speaking as someone who's got two children myself. It's very exciting that she can now join the mummy ranks.
"I'm so thrilled for them that they've got such a lovely baby. I've spoken to her already and she and the baby are doing good."
Mary King, another member of the same British Olympic eventing team, said: "I'd not heard the news until now but I'm over the moon for her.
"It's brilliant - a future Olympian no doubt in years to come. She has the right lineage to be one. I had my children while eventing and came back to competing so hopefully Zara will recover quickly and we will see her back eventing very soon."
The new family of three are believed to have headed to their home on the Princess Royal's Gatcombe Park estate near Stroud, Gloucestershire.
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.
William Hill spokesman Rupert Adams said: "The money suggests that Zara and Mike certainly have Emma on a shortlist and it would come as no surprise if it gets the nod."
More than 70% of William Hill customers predicted the couple would have a girl. Mr Adams added: "Everyone was definitely sure it was a girl. We've paid out a fairly substantial amount."
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.