Compact complete minimal immersions in R^3

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 $X:M\to {\mathbb{R}}^3$ which can be extended to a continuous map $X:\bar{M}\to {\mathbb{R}}^3,$ such that $X_{\mathrm{\mid }\mathrm{\partial }M}$ 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 ${\mathbb{R}}^3$, endowed with the topology of the Hausdorff distance.