The McLean Museum and Watt Library complex in Greenock will be closed for repairs from December.
Maintenance work includes repairs to the parapets and towers, window replacement and roof repairs and treatment for wet rot.
The council is investing £1.8m towards the refurbishment work alongside a £287,000 grant from Historic Environment Scotland.
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Plans have also been approved to move high-value and particularly fragile exhibits and artefacts to safe storage while the work is carried out.
Cllr Terry Loughran said: "Where possible as many archives, books and objects will be left on site in areas unaffected by the building work. Comprehensive measures to protect the large and heavy pieces that cannot be moved will of course be taken.
"We are anticipating the repairs will take between eighteen months and two years to complete and we are of course looking at options to provide some of the museum and library's services elsewhere during the closure."
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