Abingdon museum will continue to open seven days week - one of only a few in the world to do so - but may close on Sunday mornings.
Most museums close on Mondays but Abingdon has remained an exception because Monday is market day and the museum attracts people who visit its coffee bar and gift shop.
But the cost of the seven-day operation, long-running exhibitions and extra demands on curator Cherry Gray involving work on a possible relocation, caused the town council to reconsider the opening times.
The council has joined forces with the Vale of White Horse District Council, Oxfordshire County Council and Abingdon Performing Arts Group Enterprise in a project to turn the empty Old Gaol leisure centre into an arts and cultural centre.
A team of consultants is studying the feasibility of the project, which it is estimated would cost about £12m.
Heritage Lottery money is being sought.
Mrs Gray said she believed that future temporary exhibitions should be suspended after next spring so that she could devote more time to the Old Gaol project. She said: "The medical exhibition scheduled for next year could be cancelled but we could still have a semi-permanent exhibition on the history of Abingdon next year which could incorporate some seasonal changes."
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