Quantum mechanical boundary conditions along a timelike line, corresponding to the origin in radial coordinates, in two-dimensional dilaton gravity coupled to N matter fields, are considered. Conformal invariance and vacuum stability severely constrain the possibilities. The simplest choice found corresponds to a nonlinear Liouville-type boundary interaction. The scattering of low-energy matter off the boundary can be computed perturbatively. It is found that weak incident pulses induce damped oscillations at the boundary while large incident pulses produce black holes. The response of the boundary to such pulses is semiclassically characterized by a second order, nonlinear ordinary differential equation which is analyzed numerically.