  We present a number of qualitative arguments which strongly suggest that
extremal dyonic black holes in the 4--dimensional low energy, classical field
theory limit of toroidally compactified heterotic string theory represent
largely degenerate classes of states in the quantum theory. We propose a simple
expression for the full non--perturbative degeneracy, which contains no free
continuous parameters and reproduces the Bekenstein-Hawking $S={A\over 4G_N}$
in the large-area limit (with the ${1\over 4}$ arising from microphysics). We
sketch the elements of a physical picture leading to this expression: the holes
support much hair, such that in counting it we are led to an effective string
theory, and a matching condition whose solutions give the degeneracy.
