Zara gave birth at Gloucestershire Royal Hospital to her first child, which weighed 7lbs 12oz.
Father Mike Tindall, the former England rugby star, was present at the birth of his daughter, who is 16th in line to the throne.
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."
Asked for the Prime Minister's response to news of the birth, David Cameron's official spokesman said: "He passes on his many congratulations to the happy couple and to Her Majesty the Queen."
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.
The silver medal-winning Olympian, who is one of Britain's leading equestrian riders, 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 follows Zara's brother Peter Phillips' two children and Prince George to become the Queen and the Duke of Edinburgh's fourth great grandchild.