  We compute the one-loop contribution to the free energy in eleven-dimensional
supergravity, with the eleventh dimension compactified on a circle of radius
$R_{11}$. We find a finite result, which, in a small radius expansion, has the
form of the type IIA supergravity free energy plus non-perturbative corrections
in the string coupling $g_A$, whose coefficients we determine. We then study
type IIA superstring theory at finite temperature in the strong coupling regime
by considering M-theory on $R^9\times T^2$, one of the sides of the torus being
the euclidean time direction, where fermions obey antiperiodic boundary
conditions. We find that a certain winding membrane state becomes tachyonic
above some critical temperature, which depends on $g_A$. At weak coupling, it
coincides with the Hagedorn temperature, at large coupling it becomes $T_{\rm
cr} \cong 0.31 l_P^ {-1} $ (so it is very small in string units).
