  We prove that for any open Riemann surface $M$ and any non constant harmonic
function $h:M \to \mathbb{R},$ there exists a complete conformal minimal
immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with
$h.$
  As a consequence, complete minimal surfaces with arbitrary conformal
structure and whose Gauss map misses two points are constructed.
