  For appropriately values of $H$, we obtain an area estimate for a complete
non-compact $H$-surface of finite topology and finite area, embedded in a
three-manifold of negative curvature. Moreover, in the case of equality and
under additional assumptions, we prove that a neighbourhood of the mean convex
side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also,
we show by an counter-example that although that area estimate holds for
minimal surfaces, one does not have rigidity for equality in this case.
