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We obtain a height estimate concerning to a compact spacelike hypersurface Σn immersed with constant mean curvature H in the anti-de Sitter space ℍ1n+1, when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space ℍn. Our estimate depends only on the value of H and on the geometry of ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of ℍ1n+1 and nonexistence results in connection with such types of hypersurfaces.