  We define a mass-type invariant for asymptotically hyperbolic manifolds with
a noncompact boundary which are modelled at infinity on the hyperbolic
half-space and prove a sharp positive mass inequality in the spin case under
suitable dominant energy conditions. As an application we show that any such
manifold which is Einstein and either has a totally geodesic boundary or is
conformally compact and has a mean convex boundary is isometric to the
hyperbolic half-space.
