The work will be largely funded by a grant of almost £1.5m from the Arts Council.
Brighton & Hove City Council is also providing £500,000 towards the project.
Hedley Swain, chief executive of Brighton & Hove Museums, said: “We’re sorry the museum will close during this period as we know visitors will be disappointed.
“The work on the roof is essential to protect visitors and staff, our collections and the building.
“It needs to be carried out at a time when the weather is expected to be dry, to reduce the risk of water damage.”