“The project will safeguard our museum and globally important collection so that future generations and communities can experience and enjoy the incredible story of slate,” said chief executive Jane Richardson.
“This redevelopment will transform how we tell the story of slate, making our visitor experience even more exciting.”
The work involves repairs to the building housing the museum, as well as better access for people with disabilities and a new learning centre, shop and cafe.
It opened in 1972 and is in a Grade I listed complex of Victorian buildings which used to be the workshops for the nearby Dinorwig slate quarry.
More than 8,000 items from the museum are being put into temporary storage near Bangor. Staff have been working for weeks to record everything before it is moved so the builders can begin work.
The slate landscapes of Wales were awarded world heritage status by Unesco (United Nations Educational, Scientific and Cultural Organization) in 2021, joining attractions like the Taj Mahal, Machu Picchu and the pyramids of Egypt.