Let and be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics and respectively, and assume that is a smooth metric with bounded Gauss curvature and finite area. The paper establishes the existence of homeomorphisms between and that minimize the Dirichlet energy. Among all homeomorphisms between doubly connected Riemann surfaces such that there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.