  We have searched for two-body charmless hadronic decays of $B$ mesons. Final
states include $\pi\pi$, $K \pi$, and $KK$ with both charged and neutral kaons
and pions; $\pi\rho$, $K \rho$, and $K^*\pi$; and $K\phi$, $ K^*\phi$, and
$\phi\phi$. The data used in this analysis consist of 2.6~million
$B\bar{B}$~pairs produced at the $\Upsilon(4S)$ taken with the CLEO-II detector
at the Cornell Electron Storage Ring (CESR). We measure the branching fraction
of the sum of $B^0 \rightarrow \pi^+\pi^-$ and $B^0 \rightarrow K^+\pi^-$ to be
$(1.8^{+0.6+0.2}_{-0.5-0.3}\pm0.2) \times 10^{-5}$. In addition, we place upper
limits on individual branching fractions in the range from $10^{-4}$ to
$10^{-6}$.
