  This work resolves a longstanding tension between the physically-expected
stability of the microcanonical ensemble for gravitating systems and the fact
that the known negative mode of the asymptotically flat Schwarzschild black
hole decays too rapidly at infinity to affect the ADM energy boundary term at
infinity. The key to our study is that we fix an appropriate {\it off-shell}
notion of energy, which we obtain by constructing the microcanonical partition
function as an integral transform of the canonical partition function. After
applying the rule-of-thumb for Wick rotations from our recent companion paper
to deal with the conformal mode problem of Euclidean gravity, we find a
positive definite action for linear perturbations about any Euclidean
Schwarzchild (-AdS) black hole. Most of our work is done in a cavity with
reflecting boundary conditions, but the cavity wall can be removed by taking an
appropriate limit.
