  In this paper we find, for any arbitrary finite topological type, a compact
Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the
fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$
which can be extended to a continuous map $X:\bar{M}\to\R^3,$ such that
$X_{|\partial M}$ is an embedding and the Hausdorff dimension of $X(\partial
M)$ is $1.$
  We also prove that complete minimal surfaces are dense in the space of
minimal surfaces spanning a finite set of closed curves in $\R^3$, endowed with
the topology of the Hausdorff distance.
