An investigation is under way after staff discovered the problem, with a temporary service now operating from a mobile library parked on Dean Bank Lane.
Councillor Richard Lewis, Culture and Leisure Convener, said: “Once library staff became aware of the water damage and the need to close the library they acted swiftly to put in place plans for an emergency service for Stockbridge residents.
“The mobile library service is now operating. People can also renew their books online and I want to reassure everyone that we are progressing repairs to get the library open as soon as possible.
“Investigations are continuing to find out what caused the water damage and led to the library closing for this urgent repair work.”
Library staff are also based at the offices of charity Life Care in Cheyne Street for borrowers who wish to return books.