  We describe searches for B meson decays to the charmless vector-vector final
states omega omega and omega phi with 471 x 10^6 B Bbar pairs produced in e+ e-
annihilation at sqrt(s) = 10.58 GeV using the BABAR detector at the PEP-II
collider at the SLAC National Accelerator Laboratory. We measure the branching
fraction B(B0 --> omega omega) = (1.2 +- 0.3 +0.3-0.2) x 10^-6, where the first
uncertainty is statistical and the second is systematic, corresponding to a
significance of 4.4 standard deviations. We also determine the upper limit B(B0
--> omega phi) < 0.7 x 10^-6 at 90% confidence level. These measurements
provide the first evidence for the decay B0 --> omega omega, and an improvement
of the upper limit for the decay B0 --> omega phi.
