  Given an unbounded domain $\Omega$ of a Hadamard manifold $M$, it makes sense
to consider the problem of finding minimal graphs with prescribed continuous
data on its cone-topology-boundary, i.e., on its ordinary boundary together
with its asymptotic boundary. In this article it is proved that under the
hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet
problem is solvable if $\Omega$ satisfies certain convexity condition at
infinity and if $\partial \Omega$ is mean convex. We also prove that mean
convexity of $\partial \Omega$ is a necessary condition, extending to unbounded
domains some results that are valid on bounded ones.
