The Princess of Wales was whisked to the clinic on Tuesday for the planned procedure and underwent successful abdominal surgery.
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The Royal mum-of-three is expected to remain in hospital for 10 to 14 days before returning home to continue her recovery, said Kensington Palace.
Officials have not revealed the nature of the surgery due to Kate’s “medical privacy” but a source said she was doing well.
Kate is likely to spend between two and three months recuperating and is not expected to return to public duties until after Easter.
The Waleses’ three children George, ten, Charlotte, eight, and Louis, five, were all at Lambrook School in Berkshire as usual as the family keep things as normal as possible.
Future king William has been doing the school run in between trips to see his wife.
He is expected to be a regular visitor, travelling in 25 miles from their home, Adelaide Cottage in Windsor, Berks.
Wills will cut back on his number of public engagements to take on more parental duties.
The caring Prince was at the Princess of Wales’s bedside before and after Tuesday;s surgery on her abdomen.
But it has forced Kate and William to postpone top-secret plans to visit British troops stationed abroad.
An official visit to Italy in February has been called off and William will not travel alone.
Kate’s major projects launched last year, including her Shaping Us initiative for under-fives, will take a back seat while she recovers.
The Prince of Wales will also not undertake official duties while his wife is in hospital and during the immediate period following her return home.
He and Kate had been expected to feature heavily at events marking Mental Health Awareness Week next month.
A statement from Kensington Palace said: “Her Royal Highness The Princess of Wales was admitted to The London Clinic yesterday for planned abdominal surgery.
“The surgery was successful and it is expected that she will remain in hospital for ten to 14 days, before returning home to continue her recovery.
“Based on the current medical advice, she is unlikely to return to public duties until after Easter.
“The Princess of Wales appreciates the interest this statement will generate. She hopes the public will understand her desire to maintain as much normality for her children as possible; and her wish that her personal medical information remains private.
“Kensington Palace will, therefore, only provide updates on Her Royal Highness’s progress when there is significant new information to share.
“The Princess of Wales wishes to apologise to all those concerned for the fact that she has to postpone her upcoming engagements.
“She looks forward to reinstating as many as possible, as soon as possible.”
Less than 90 minutes later in a second royal shock, Buckingham Palace announced yesterday that the King would need an operation on an enlarged prostate.
It is understood that William and Kate have been in touch with Charles and Queen Camilla.