Normal subgroups of groups which split over the infinite cyclic group

Let G be either a free product with amalgamation A *(C) B or an HNN group A *(C), where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of inf inite indices. We show that if G contains a finitely generated normal subgr oup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.