Marne is an unincorporated community and census-designated place in Licking County, Ohio, United States. As of the 2020 census it had a population of 772. Marne is in eastern Licking County, in the eastern part of Madison Township. Ohio State Route 16 forms the southern edge of the community, leading west 5 miles to Newark, the Licking county seat, …Marne is an unincorporated community and census-designated place in Licking County, Ohio, United States. As of the 2020 census it had a population of 772. Marne is in eastern Licking County, in the eastern part of Madison Township. Ohio State Route 16 forms the southern edge of the community, leading west 5 miles to Newark, the Licking county seat, and northeast 32 miles to Coshocton. Marne is in the valley of the Licking River, which passes to the south of State Route 16. The Licking is an east-flowing tributary of the Muskingum River and part of the Ohio River watershed.