Black hole entropy from horizon conformal field theory

String theory and "quantum geometry" have recently offered independent stat istical mechanical explanations of black hole thermodynamics. But these suc cesses raise a new problem: why should models with such different microscop ic degrees of freedom yield identical results? I propose that the asymptoti c behavior of the density of states at a black hole horizon may be determin ed by an underlying symmetry inherited from classical general relativity, i ndependent of the details of quantum gravity. I offer evidence that a two-d imensional conformal symmetry at the horizon, with a classical central exte nsion, may provide the needed behavior.