Reference request: Fuchsian Model of compact Riemann surface

Question

Let $S$ be a compact Riemann surface with genus $\geq$ 2. Then by the Uniformization theorem it has universal covering space the upper half-plane $\mathbb{H}$(up to conformal equivalence). Now I read that we can express $S$ as $\mathbb{H}/\Gamma$, where $\Gamma$ is a stricly hyperbolic Fuchsian group. The statement sounds reasonable, but I want to have a actual proof of that. Can someone suggest a reference for that? Thanks a lot.