  Understanding how to prepare and count black hole micro-states by using the
gravitational path integral is one of the most important problems in quantum
gravity. Nevertheless, a state-by-state count of black hole microstates is
difficult because the apparent number of degrees of freedom available in the
gravitational effective theory can vastly exceed the entropy of the black hole,
even in the special case of BPS black holes. In this paper, we show that we can
use the gravitational path integral to prepare a basis for the Hilbert space of
all BPS black hole microstates. We find that the dimension of this Hilbert
space computed by an explicit state count is in complete agreement with the
degeneracy obtained from the Gibbons-Hawking prescription. Specifically, this
match includes all non-perturbative corrections in $1/G_N$. Such corrections
are, in turn, necessary in order for this degeneracy of BPS states to match the
non-perturbative terms in the $1/G_N$ expansion in the string theory count of
such microstates.
