  We study and solve the Dirichlet problem for graphs of prescribed mean
curvature in $\mathbb R^{n+1}$ over general domains $\Omega$ without requiring
a mean convexity assumption. By using pieces of nodoids as barriers we first
give sufficient conditions for the solvability in case of zero boundary values.
Applying a result by Schulz and Williams we can then also solve the Dirichlet
problem for boundary values satisfying a Lipschitz condition.
