Given an irreducible module for the affine Hecke algebraH n of type A, we consider its restriction toH n−1. We prove that the socle of restriction is multiplicity free and moreover that the summands lie in distinct blocks. This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofH n . The result generalizes and implies the classical “branching rules” that describe the restriction of an irreducible representation of the symmetric groupS n toS n−1.