Determining the stability of solutions is an important aspect of the analysis of an applied model, because it is typically the stable solutions that are observed in practice. A technique for determining nonlinear stability is presented in the context of scalar viscous conservation laws. In the associated linear operator, there is no spectral gap between the neutral and decaying modes, which prevents the application of standard methods in stability analysis. To overcome this difficultly, scaling variables are used to open up a gap in the spectrum, thus allowing for the use of invariant manifold theory. Detailed information is obtained about the temporal decay rate of perturbations and the geometry that governs their evolution.