  We prove the property of stochastic stability previously introduced as a
consequence of the (unproved) continuity hypothesis in the temperature of the
spin-glass quenched state. We show that stochastic stability holds in
beta-average for both the Sherrington-Kirkpatrick model in terms of the square
of the overlap function and for the Edwards-Anderson model in terms of the bond
overlap. We show that the volume rate at which the property is reached in the
thermodynamic limit is V^{-1}. As a byproduct we show that the stochastic
stability identities coincide with those obtained with a different method by
Ghirlanda and Guerra when applyed to the thermal fluctuations only.
