We augment the standard model gauge groups by an additional $U(1)_X$ group with the corresponding gauge boson, $Z^\prime$, in the context of the type-I seesaw. After breaking $U(1)_X$ with vacuum expectation values of even charge, the lightest of particles, one of the heavy neutrinos, with odd $U(1)_X $ charge is a dark matter candidate. We first find the minimum anomaly-free particle content and then discuss the light neutrino mass spectrum and mixing matrix, which in this model consists of radiative corrections and tree-level contributions from the right-handed neutrinos. Second, in terms of the relic density, we investigate the interplay between annihilation processes via the $Z^\prime$ exchange and co-annihilation processes, including experimental constraints, and reproducing the observed neutrino mixing angles and mass-squared differences.