2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/218971In 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.16 pages. Main theorem improved. To appear in Trans. Amer. Math. SocDifferential Geometry53A10; 53C42; 49Q05; 49Q10text