We present evidence for a new type of giant magnetoresistance in (111) cobalt-copper superlattices with atomically smooth interfaces. We propose that the lowered dimensionality of the structure leads to an enhancement of the scattering of conduction electrons from paramagnetic interfaces obeying a Langevin-like saturation at very high fields, well beyond the switching field of the Co layers. The findings help to explain similarities in magnetotransport behavior with recently reported granular systems as well as differences with antiferromagnetically coupled multilayers