An efficient nondegenerate four-wave-mixing process in a highly birefringent optical fiber pumped by a dye laser is reported. The output beam from a dye laser (at ~610 nm) was found to mix with the accompanying superfluorescent light (from 575 to 600 nm) in a birefringent fiber to generate a distinct frequency-shifted beam, which could be Stokes or anti-Stokes radiation, depending on which mode of the fiber was excited. The frequency shift was comparable with that observed in the well-known pump-divided degenerate process, as confirmed by the theory.