Edward is flying back from a state visit to Mauritius

Sophie's father, Christopher Rhys-Jones, gave the upbeat prognosis after visiting his daughter in Frimley Park Hospital in Surrey hours after the birth late on Saturday.

His so far unnamed granddaughter is being cared for at the neo-natal unit at St George's Hospital in Tooting, south London, where she was transferred after being delivered by Caesarean section.

The Earl of Wessex is due to arrive back in London on Sunday evening after flying back from a state visit to Mauritius to be with his wife and daughter.

The Queen was said to be "pleased to hear that they are doing well", Buckingham Palace said.

She is being kept regularly informed of the progress of both mother and baby, who was born a month early at 2332 GMT on Saturday weighing 4lb 9oz.

Hospital stay

Doctors said both were stable, but the baby was transferred to the neo-natal unit "as a precaution". She is the first grandchild of the Queen to be born in an NHS hospital.

Sophie will remain in hospital "for the next five days", Buckingham Palace said.

The parents of the countess brought her flowers

Mr Rhys-Jones, with his wife Mary, spent more than an hour with Sophie and said afterwards: "Sophie is absolutely fine. She's heavily sedated and rather sleepy but she is perfectly OK. The baby is also fine."

The Rhys-Joneses then drove to Tooting to see their grandchild.

The pregnancy had been progressing well and royal gynaecologist Marcus Setchell had expected to deliver the child early in December.

But Sophie, who had been due to travel to London for Sunday's Remembrance Day service and parade at the Cenotaph with other members of the Royal Family, was taken ill at her Bagshot Park home on Saturday night.

Earl returns

Mr Setchell was present at the nearby hospital for the emergency operation, which was carried out by surgeon Sukhpal Singh with gynaecologist Anne Deans and midwife Adrienne Price.

SUCCESSION TO THE THRONE 1: Charles, the Prince of Wales 2: Prince William 3: Prince Harry 4: Andrew, the Duke of York 5: Princess Beatrice 6: Princess Eugenie 7: Edward, the Earl of Wessex 8: Edward and Sophie's child

Prince Charles, attending a Remembrance ceremony in Oman, told reporters he was "delighted" at the news of the birth of his niece.

The child, the first for the Queen's youngest son and his wife, is the Queen's seventh grandchild and eighth in line to the throne.

When Edward and Sophie married in 1999, it was decided, with the Queen's agreement, that any children they had would not be called His or Her Royal Highness.

Instead, the couple's baby girl will take the title of Lady followed by her chosen Christian name and then Windsor or possibly Mountbatten-Windsor.

It is understood they have yet to decide on the baby's name. Despite routine scans, they had not asked to be told the gender of the baby.

Serious condition

The birth follows a traumatic failed pregnancy in December 2001 when Sophie was airlifted to hospital after suffering an ectopic pregnancy.

Surgeons operated to remove the foetus from her fallopian tube after she was rushed to King Edward VII Hospital by air ambulance.

Doctors said that if the condition had continued undiagnosed, it could have been life-threatening.

This time the countess initially suspended official engagements in the early stages of the pregnancy, on doctors' advice, but had resumed duties by June.

Three days ago she opened new offices for the ChildLine charity in London.