  Due to the failure of thermodynamics for low temperature near-extremal black
holes, it has long been conjectured that a "thermodynamic mass gap" exists
between an extremal black hole and the lightest near-extremal state. For
non-supersymmetric near-extremal black holes in Einstein gravity, with an
AdS$_2$ throat, no such gap was found. Rather, at that energy scale, the
spectrum exhibits a continuum of states, up to non-perturbative corrections. In
this paper, we compute the partition function of near-BPS black holes in
supergravity where the emergent, broken, symmetry is $PSU(1,1|2)$. To reliably
compute this partition function, we show that the gravitational path integral
can be reduced to that of a $\mathcal N=4$ supersymmetric extension of the
Schwarzian theory, which we define and exactly quantize. In contrast to the
non-supersymmetric case, we find that black holes in supergravity have a mass
gap and a large extremal black hole degeneracy consistent with the
Bekenstein-Hawking area. Our results verify several string theory conjectures,
concerning the scale of the mass gap and the counting of extremal micro-states.
