We describe the branch locus of a proper holomorphic mapping between two sm
oothly bounded pseudoconvex domains of finite type In C-2 under the assumpt
ion that the first domain admits a transversal holomorphic action of the un
it circle. As an application we show that any proper holomorphic self-mappi
ng of a smoothly bounded pseudoconvex complete circular domain of finite ty
pe in C-2 is biholomorphic.