Let $G$ be a finite subgroup of $GL(n,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $K$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of $D$ as a quotient of the skew group ring A=S*G by the idempotent $e$ corresponding to the trivial representation. We will explain how this can be seen in some sense as a McKay correspondence for reflection groups. This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.