  Using quasiconformal mappings, we prove that any Riemann surface of finite
connectivity and finite genus is conformally equivalent to an intrinsic circle
domain U in a compact Riemann surface S. This means that each connected
component B of S \ U is either a point or a closed geometric disc with respect
to the complete constant curvature conformal metric of the Riemann surface (U
union B). Moreover the pair (U,S) is unique up to conformal isomorphisms. We
give a generalization to countably infinite connectivity. Finally we show how
one can compute numerical approximations to intrinsic circle domains using
circle packings and conformal welding.
