When using triangle meshes in numerical simulations or other sophisticated downstream applications, we have to guarantee that no degenerate faces are present since they have, e.g., no well defined normal vectors. In this paper we present a simple but effective algorithm to remove such artifacts from a given triangle mesh. The central problem is to make this algorithm numerically robust because degenerate triangles are usually the source for all kinds of numerical instabilities. Our algorithm is based on a slicing technique that cuts a set of planes through the given polygonal model. The mesh slicing operator only uses numerically stable predicates and therefore is able to split faces in a controlled manner. In combination with a custom tailored mesh decimation scheme we are able to remove the degenerate faces from meshes like those typically generated by tesselation units in CAD systems.