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The baby weighs just 4lbs 9oz (2 kg) and was delivered by emergency Caesarian section at Frimley Park hospital in Surrey at 2332 GMT.
She was born without her father present: Prince Edward, the Earl of Wessex, was on an official visit to Mauritius. He is flying home to be with his wife and daughter.
The countess, 38, called an ambulance after being taken ill with cramps at her home in Bagshot Park.
Both mother and baby are said to be stable, but a Frimley hospital spokesman said the baby had been transferred to a neo-natal unit at St George's Hospital in Tooting, London, "as a precaution".
Sophie would remain in hospital "for the foreseeable future", he said.
The pregnancy had been progressing well and royal gynaecologist Marcus Setchell had expected to deliver the child early in December.
Mr Setchell was present at Frimley Park hospital for the emergency operation.
The baby is the first grandchild of the Queen to be born in an NHS hospital.
The child is the first for the Queen's youngest son and his wife, and is the Queen's seventh grandchild. She will be eighth in line to the throne.
When Edward and Sophie married in 1999, it was decided that any children they had would not be called His or Her Royal Highness, in an attempt to shield them from media attention.
The baby is likely to take the title of Lady, followed by a given name to be decided, and then Windsor or possibly Mountbatten-Windsor.
The Queen is being kept informed of events, and Buckingham Palace have said she is pleased to hear that they are doing well.
The birth follows a traumatic failed pregnancy in December 2001, when Sophie was airlifted to hospital after suffering an ectopic pregnancy.
The couple have faced controversy and media criticism as they struggled to combine high-profile jobs in television and public relations with royal duties since they married in 1999.
In March last year the couple announced they would leave their separate companies to devote themselves exclusively to royal duties.
Prince Edward arrived back in London the following evening to see his wife and shortly afterwards his new baby daughter.
Sophie and her baby were reunited six days later. Sophie left hospital on 19 November, and her daughter was finally allowed home four days later.
It emerged that the countess had had to wait 30 minutes for an ambulance, after a mix-up by Surrey police meant that officers were despatched to Sophie's home, but not an ambulance.
The force apologised to Sophie, and the duty inspector in charge of the force's control room at the time was temporarily transferred to a desk job.
The baby's name was not announced until 27 November. She was christened Louise Alice Elizabeth Mary Mountbatten-Windsor on 24 April 2004, and will be styled Lady Louise Windsor.
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