We present evidence for the flavor-changing neutral current decay B-->K*l+l- and a measurement of the branching fraction for the related process B-->K l+l-, where l+l- is either an epsilon+epsilon- or a mu+mu- pair. These decays are highly suppressed in the standard model, and they are sensitive to contributions from new particles in the intermediate state. The data sample comprises 123 x 10(6) Upsilon(4S)-->B(-)B decays collected with the BABAR detector at the SLAC PEP-II epsilon+epsilon- storage ring. Averaging over K(*) isospin and lepton flavor, we obtain the branching fractions B(B-->Kl+l-)=(0.65(+0.14)(-0.13)+/-0.04)x10(-6) and B(B-->K*l+l-)=(0.88(+0.33)(-0.29)+/-0.10)x10(-6), where the uncertainties are statistical and systematic, respectively. The significance of the B-->Kl+l- signal is over 8sigma, while for B-->K*l+l- it is 3.3sigma.