Let M and N be doubly connected Riemann surfaces with \({\mathscr {C}}^{1,\alpha }\) boundaries and with nonvanishing conformal metrics \(\sigma \) and \(\wp \) respectively, and assume that \(\wp \) is a smooth metric with bounded Gauss curvature \({\mathcal {K}}\) and finite area. Assume that \({\mathcal {H}}^\wp (M, N)\) is the class of all \({\mathscr {W}}^{1,2}\) homeomorphisms between M and N and assume that \({\mathcal {E}}^\wp : {\mathcal {H}}^\wp (M, N)\rightarrow {\mathbf {R}}\) is the Dirichlet-energy functional, where \(\overline{{\mathcal {H}}}^\wp (M, N)\) is the closure of \({{\mathcal {H}}}^\wp (M, N)\) in \({\mathscr {W}}^{1,2}(M,N)\). By using a result of Iwaniec, Kovalev and Onninen in Iwaniec et al. (Duke Math J 162(4):643–672, 2013) that the minimizer, is locally Lipschitz, we prove that the minimizer, of the energy functional \({\mathcal {E}}^\wp \), which is not a diffeomorphism in general, is a globally Lipschitz mapping of M onto N. Note that, this result is new also for flat Riemann surfaces, i.e. for the planar domains furnished with the Euclidean metric.