  Motivated by the recent conjecture of Ooguri, Strominger and Vafa, we compute
the semi-canonical partition function of BPS black holes in N=4 and N=8 string
theories, to all orders in perturbation theory. Not only are the black hole
partition functions surprisingly simple; they capture the full topological
string amplitudes, as expected from the OSV conjecture. The agreement is not
perfect, however, as there are differences between the black hole and
topological string partition functions even at the perturbative level. We
propose a minimal modification of the OSV conjecture, in which these
differences are understood as a nontrivial measure factor for the topological
string.
