We study the homogeneous sector of the RST model describing the gravitational
dynamics, including back-reaction, of radiating 2-d black holes. We find the
exact solutions both in conformal gauge and in time-parametrized form, isolate
the black hole sector of the classical phase space and quantize the near
singularity dynamics in conformal gauge. We show that different choices of
measure and different self-adjoint extensions can lead to inequivalent quantum
theories, all of which resolve the singularity. For a specific range of
extension parameters, the Hamiltonian spectrum admits bound states that
correspond physically to stable remnants. Finally, we argue that our work
provides a good starting point for quantization of the full homogeneous theory
using both reduced and Dirac quantization