  In this paper, we show that if $R$ is a compact Riemann surface and
$M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of
countably many pairwise disjoint smoothly bounded closed discs $D_i$, then $M$
is the complex structure of a complete bounded minimal surface in $\mathbb
R^3$. We prove that there is a complete conformal minimal immersion
$X:M\to\mathbb R^3$ extending to a continuous map $X:\overline M\to\mathbb R^3$
such that $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan
curves. This extends a recent result for bordered Riemann surfaces.
