We show magnetoresistance in excess of 1000% in trilayers containing highly spin-polarized La0.7Ca0.3MnO3 and high-Tc superconducting YBa2Cu3O7. This large magnetoresistance is reminiscent of the giant magnetoresistance (GMR) in metallic superlattices but with much larger values, and originates at spin imbalance due to the injection of spin-polarized carriers. Furthermore, in contrast to ordinary GMR, the magnetoresistance is intimately related to the superconductivity in the YBa2Cu3O7 layer and vanishes in the normal state. This result, aside from its fundamental importance, may be of interest for the design of novel spintronic devices based on ferromagnet/superconductor structures.