We prove that a Dedekind domain R , graded by a nontrivial torsionfree abelian group, is either a twisted group ring k'[G] or a polynomial ring k[X], where k is a field and G is an abelian torsionfree rank one group. It follows that R is a Dedekind domain if and only if R is a principal ideal domain. We also investigate the case when R is graded by an arbitrary nontrivial torsionfree monoid.