On sandwiched singularities and complete ideals

Given a complete ideal I in a two-dimensional normal complete local -algebra having a rational singularity, we prove that there is a bijection between the set of complete ideals of codimension one contained in I and the set of points in the exceptional locus of the surface X=BlI(R). As a consequence, in the case the ring R is regular, we are able to read from the Enriques diagram of the cluster of base points of I the number of singularities on X as well as their fundamental cycles and multiplicities.