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We have searched for two-body charmless decays of B mesons to purely hadronic exclusive final states including ω or φ mesons using data collected with the CLEO II detector. With this sample of 6.6 X 106 B mesons we observe a signal for the ωK+ final state, and measure a branching fraction of Ɓ(B+→ωK+) = (1.5-0.6+0.7 ± 0.2) X 10-5. We also observe some evidence for the φK* final state, and upper limits are given for 22 other decay modes. These results provide the opportunity for studies of theoretical models and physical parameters.