We report results on the appearance of temporally stable Stokes emission in stimulated Brillouin scattering (SBS) excited by continuous-wave pump radiation in optical fiber. With increasing pump strength the stable component emerges from stochastic emission slightly above the SBS threshold to become the dominant contribution, independent of fiber characteristics. These findings are shown to be a manifestation of spectral self-phase conjugation, providing what is to our knowledge the first experimental evidence of this phenomenon in optics.