High-efficiency mutually pumped phase conjugation (MPPC) has been realized in a BaTiO$_3$:Ce crystal in the bridge geometry. The laser source is a dye laser operating from 580 to 650 nm. We have observed that, with a beam ratio of 1, self-pumped phase conjugation (SPPC) appears and competes with MPPC at short wavelengths while it disappears at long wavelengths. When the beam ratio is slightly higher than 1, SPPC no longer occurs. Phase conjugate reflectivity near 500% has been obtained at a beam ratio of 70. Our experiments demonstrate that Ce-doped BaTiO$_3$crystals, which have already been found efficient in SPPC experiments, are also efficient for the generation of MPPC.