  We study the rare B meson decays B{+-,0} --> J/\psi\ K^+ K^- K{+-,0}, B{+-,0}
--> J/\psi\ \phi\ K{+-,0}, and search for B0 --> J/\psi\ \phi, using 469
million BBbar events collected at the Upsilon(4S) resonance with the BABAR
detector at the PEP-II e+e- asymmetric-energy collider. We present new
measurements of branching fractions and a study of the J/\psi\phi mass
distribution in search of new charmonium-like states. In addition, we search
for the decay B0 --> J/\psi\ \phi, and find no evidence of a signal.
