Let \(M^n\) be a noncompact closed hypersurface in \(n+1\)-dimensional hyperbolic space \({\mathbf H}^{n+1}\) which bounds a convex set. If furthermore \(M^n\) has nonnegative curvature everywhere and one mean curvature of \(M^n\) is constant: \(H_r = c\) for some \(r\) with \(1\leq r\leq\frac{2}{3}(n+1)\) then \(M^n\) is either a horosphere or a geodesic cylinder. If \(\frac{2}{3}(n+1)< r \leq n\) and moreover \(c\) lies in a specified intervall the same holds true. The (extensive) proof of this theorem is given in the paper.