The farmer co-op is part-funding the site by selling its Campbeltown Creamery to Tesco, which plans a superstore in its place.
Up to £3.9m will come from a Scottish government grant for the new site, also in Campbeltown.
Campbeltown Creamery was "struggling to cope with modern demands", First Milk said. The new facility would provide increased production, better processing and the treatment of whey as a waste product.
"The creamery and dairy farming community on Kintyre are mutually dependent, and this new investment secures the future of the dairy industry in this part of Scotland," said CEO Peter Humphreys.
A public consultation with Tesco will now follow and planning permission for both sites will be needed before work starts.