Kate, 42, was admitted to The London Clinic on Tuesday (16 January) for a planned abdominal procedure, Kensington Palace announced.
The future Queen will likely spend 10 to 14 more days in hospital before continuing her recovery at home.
Kate is likely to spend between two and three months recuperating and is not expected to resume public duties until after Easter.
The Prince of Wales will not undertake official duties while his wife is in hospital and during the immediate period following her return home.
Prince William is expected to combine being at his wife's side with caring for their children Prince George, Princess Charlotte and Princes Louis.
The heir to the throne has postponed several upcoming engagements.
Palace representatives would not confirm what the princess was being treated for but said the condition was non-cancerous.
A spokesperson said: “The Princess of Wales appreciates the interest this statement will generate.
“She hopes that 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.