Knightswood library will shut on July 29 for a planned programme of upgrades and repairs costing around £300,000.
As well as removing the asbestos, Glasgow Life also plans to fit new windows and automatic doors.
It is hoped the West End building will reopen in November.
Library users will be able to return any books they have borrowed by using a temporary returns box which will be located in the foyer of Knightswood community centre.
Borrowers can use any Glasgow library while the work is being carried out with Drumchapel, Anniesland, Whiteinch and Partick being the nearest alternatives.
A Glasgow Life spokesman said: "Glasgow libraries play an important role in every community and this investment will ensure Knightswood Library will continue to serve local people whether they want to borrow books, use the internet or join in the classes and sessions.
"This is a planned pro- gramme of works which will remove the asbestos from the building safely and ensure it can never become an issue for people who use Knightswood library."
In June, Woodside library closed for nine weeks to allow roof repairs to be carried out.
The dome is leaking and £40,000 of work needs to be done. During the closure parts of the interior will be painted.
The nearest alternatives are Hillhead in Byres Road, the Mitchell in North Street and Maryhill library in Maryhill Road.
The spokesman said: "The building's dome is in need of some repairs which will be undertaken in the coming weeks, alongside some redecoration of the interior."
Work will start later this year on a £2million upgrade of Mitchell Library to help make it the safest possible home for its collection.
The 30-year-old fire detection system will be replaced, lighting improved and new lifts installed.
Parts of the library will be closed on a phased basis to ensure the majority of the building remains open to the public.
Work, starting on the upper floors, will be completed next year.
The Mitchell has a large public reference library with more than 1.2million volumes, making it one of Europe's largest public libraries.