Search for the Rare Decays B --> Kl^+l^- and B --> K^*l^+l^-

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