  We have studied two-body charmless decays of the $B$ meson into the final
states $\rho^0 \rho^0$, $K^{*0} \rho^0$, $K^{*0} K^{*0}$, $K^{*0}
\bar{K^{*0}}$, $K^{*+} \rho^0$, $K^{*+} \bar{K^{*0}}$, and $K^{*+} K^{*-}$
using only decay modes with charged daughter particles. Using 9.7 million $B
\bar{B}$ pairs collected with the CLEO detector, we place 90% confidence level
upper limits on the branching fractions, $(0.46-7.0)\times 10^{-5}$, depending
on final state and polarization.
