The degenerate parabolic equation \[ u_t=\Delta\beta(u) \] is considered in the cylinder $\Omega\times(0,T)$, where $\Omega\subset{\R}^n$ is bounded, $\beta(0)=\beta'(0)=0$, and $\beta'\geq0$ (e.g., $\beta(u)=u|u|^{m-1}$, $m>1$). A fully discrete scheme is given using C0-piecewise linear finite elements in space and a backward difference scheme in time. L2-error estimates for the approximation of u and of $\beta(u)$ are proven. For that purpose we employ the fact that the free boundary has a finite speed of propagation.