  Let $D$ be a regular strictly convex bounded domain of $\r^3$, and consider a
regular Jordan curve $\Gamma \subset \partial D$. Then, for each $\epsilon>0$,
we obtain the existence of a complete proper minimal immersion $\psi_\epsilon
:\d \to D$ satisfying that the Hausdorff distance
$\delta^H(\psi_\epsilon(\partial \d), \Gamma) < \epsilon,$ where
$\psi_\epsilon(\partial \d)$ represents the limit set of the minimal disk
$\psi_\epsilon(\d).$
  This result has some interesting consequences. Among other things, we can
prove that any bounded regular domain $R$ in $\r^3$ admits a complete proper
minimal immersion $\psi: \d \longrightarrow R$.
