We describe searches for B meson decays to the charmless vector-vector final states ω ω and ω ϕ with 471 × 1 0 6 B ¯ B pairs produced in e + e − annihilation at √ 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 ( B 0 → ω ω ) = ( 1.2 ± 0.3 + 0.3 − 0.2 ) × 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 ( B 0 → ω ϕ ) < 0.7 × 10 − 6 at 90% confidence level. These measurements provide the first evidence for the decay B 0 → ω ω , and an improvement of the upper limit for the decay B 0 → ω ϕ .