Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain
Ω
\Omega
in a compact Riemann surface
S
S
. This means that each connected component
B
B
of
S
∖
Ω
S\setminus \Omega
is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface
(
Ω
∪
B
)
(\Omega \cup B)
. Moreover, the pair
(
Ω
,
S
)
(\Omega , 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.