  We present a microscopic index formula for the degeneracy of dyons in
four-dimensional N=4 string theory. This counting formula is manifestly
symmetric under the duality group, and its asymptotic growth reproduces the
macroscopic Bekenstein-Hawking entropy. We give a derivation of this result in
terms of the type II five-brane compactified on K3, by assuming that its
fluctuations are described by a closed string theory on its world-volume. We
find that the degeneracies are given in terms of the denominator of a
generalized super Kac-Moody algebra. We also discuss the correspondence of this
result with the counting of D-brane states.
