Complete minimal surfaces and harmonic functions

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 $h\mathrm{.}$ As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are constructed.