We have used element-specific hysteresis measurements, based on the x-ray magnetic circular dichroism technique, to investigate magnetic trilayer structures composed of Fe and Ni layers. Within a critical regime we have discovered a class of structures in which the exchange interaction, the mechanism responsible for the macroscopic magnetism, can become vanishingly small. The experimental observations are supported by first principles theory and are explained as arising from a cancellation of several competing magnetic interactions. Hence, we have discovered a system with a novel exchange interaction between magnetic layers in direct contact that replaces the conventional exchange interaction in ferromagnets.