  We construct harmonic diffeomorphisms from the complex plane $C$ onto any
Hadamard surface $M$ whose curvature is bounded above by a negative constant.
For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times
R$ over domains of $M$ bounded by ideal geodesic polygons and show the
existence of a sequence of minimal graphs over polygonal domains converging to
an entire minimal graph in $M\times R$ with the conformal structure of $C$.
