  In this paper, we give an optimal inequality relating the relative Yamabe
invariant of a certain compactification of a conformally compact
Poin-car{\'e}-Einstein manifold with the Yamabe invariant of its boundary at
infinity. As an application, we obtain an elementary proof of the rigidity of
the hyper-bolic space as the only conformally compact Poincar{\'e}-Einstein
manifold with the round sphere as its conformal infinity.
