We present results from a search for the flavor-changing neutral current decays B-->Kl(+)l(-) and B-->K(*)l(+)l(-), where l(+)l(-) is either an e(+)e(-) or mu(+)mu(-) pair. The data sample comprises 22.7 x 10(6) Upsilon(4S)-->B(-)B decays collected with the BABAR detector at the PEP-II B Factory. We obtain the 90% C.L. upper limits B(B-->Kl(+)l(-))<0.51 x 10(-6) and B(B-->K(*)l(+)l(-))<3.1 x 10(-6), close to standard model predictions for these branching fractions. We have also obtained limits on the lepton-family-violating decays B-->Ke+/-mu(-/+) and B-->K(*)e(+/-)mu(-/+).