In this paper we find, for any arbitrary finite topological type, a compact Riemann surface
M
,
\mathcal {M},
an open domain
M
⊂
M
M\subset \mathcal {M}
with the fixed topological type, and a conformal complete minimal immersion
X
:
M
→
R
3
X:M\to \mathbb {R}^3
which can be extended to a continuous map
X
:
M
¯
→
R
3
,
X:\overline {M}\to \mathbb {R}^3,
such that
X
|
∂
M
X_{|\partial M}
is an embedding and the Hausdorff dimension of
X
(
∂
M
)
X(\partial M)
is
1.
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
\mathbb {R}^3
, endowed with the topology of the Hausdorff distance.