  We have searched for two-body charmless decays of B mesons to purely hadronic
exclusive final states including $\omega$ or $\phi$ mesons using data collected
with the CLEO II detector. With this sample of $6.6 \times 10^6$ B mesons we
observe a signal for the $\omega K^+$ final state, and measure a branching
fraction of B($B^+ \to \omega K^+$) = 1.5 +0.7 -0.6 +-0.2 $\times 10^{-5}$. We
also observe some evidence for the $\phi K^*$ 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.
