  We present results from a search for the flavor-changing neutral current
decays $B\to K\ell^+\ell^-$ and $B\to K^*\ell^+\ell^-$, where $\ell^+\ell^-$ is
either an $e^+e^-$ or $\mu^+\mu^-$ pair. The data sample comprises $22.7\times
10^6$ $\Upsilon(4S)\to B\bar B$ decays collected with the BABAR detector at the
PEP-II $B$ Factory. We obtain the 90% C.L. upper limits ${\mathcal B}(B\to
K\ell^+\ell^-)< 0.50\times 10^{-6}$ and ${\mathcal B}(B\to
K^*\ell^+\ell^-)<2.9\times 10^{-6}$, close to Standard Model predictions for
these branching fractions. We have also obtained limits on the
lepton-family-violating decays $B\to Ke^{\pm}\mu^{\mp}$ and $B\to
K^{*}e^{\pm}\mu^{\mp}$.
