Horizon Constraints and Black Hole Entropy*

To ask a question about a black hole in quantum gravity, one must restrict initial or boundary data to ensure that a black hole is actually present. For two-dimensional dilaton gravity, and probably a much wider class of theories, I show that the imposition of a``stretched horizon'' constraint modifies the algebra of symmetries at the horizon, inducing a central term. Standard conformal field theory techniques then fix the asymptotic density of states, reproducing the Bekenstein-Hawking entropy. The states responsible for black hole entropy can thus be viewed as ``would-be gauge'' states that become physical because the symmetries are altered.