We propose a Kruzkov-type entropy condition for nonlinear degenerate parabolic equations with discointinuous coefficients. We establish L1 stability, and thus uniqueness, for weak solutions satisfying the entropy condition, provided that the flux function satisfies a so called ```crossing condition'' and the solution satisfies a technical condition regarding the existence of traces at the jump pooints in the coefficients. In some important cases, we prove the existence of traces directly from the proposed entropy condition. We show that limits generated by the Engquist-Osher finite difference scheme and front tracking (for the hyperbolic equation) satisfy the entropy condition, and are therefore unique. By combining the uniqueness and L1 stability results of this paper with previously established existence results [27, 28], we show that the initial value problem studied herein is well-posed in some important cases. Our class of equations contains conservation laws with discontinuous coefficients as well as a certain type of singular source term.