  The count of microstates for supersymmetric black holes is typically obtained
from a supersymmetric index in weakly-coupled string theory. We find the
saddles in the gravitational path integral corresponding to this index in a
general theory of $N=2$ supergravity in asymptotically flat space. This saddle
exhibits a new attractor mechanism which explains the agreement between the
string theory index and the macroscopic entropy. These saddles are smooth,
complex Euclidean spinning black holes that are supersymmetric but not
extremal, i.e., they are formally finite-temperature solutions. With this new
mechanism, the scalars and the electromagnetic fields get attracted to
temperature- and moduli-independent values at the north and south poles of the
rotating black hole, although they vary along the Euclidean horizon in a
non-universal way. Further, although the area and the spin of the black hole
depend non-trivially on the temperature and on the moduli, the free energy is
essentially a function only of the black hole charges (apart from a trivial
dependence on the temperature and the moduli through the BPS mass), and agrees
with the string theory index.
