The one-loop contribution to the entropy of a black hole from field modes
near the horizon is computed in string theory. It is modular invariant and
ultraviolet finite. There is an infrared divergence that signifies an
instability near the event horizon of the black hole. It is due to the
exponential growth of the density of states and the associated Hagedorn
transition characteristic of string theory. It is argued that this divergence
is indicative of a tree level contribution, and the Bekenstein-Hawking-Gibbons
formula for the entropy should be understood in terms of string states stuck
near the horizon.Comment: 15 pages, harvmac, replaced with version published in Phys. Letts.
