We consider the dimer model on a bipartite periodic graph with elliptic weights introduced by Fock. The spectral curves of such models are in bijection with the set of all genus 1 Harnack curves. We prove an explicit and local expression for the two-parameter family of ergodic Gibbs measures and for the slope of the measures. This is joint work with Cédric Boutillier and David Cimasoni.