  We present the results of a search for the rare flavor-changing
neutral-current decays B --> pi l+l- (pi=pi+/-, pi0 and l = e,mu) and B0 -->
eta l+l- using a sample of e+e- --> Y(4S) BB decays decays corresponding to 428
fb^-1 of integrated luminosity collected by the BABAR detector. No significant
signal is observed, and we set an upper limit on the isospin and lepton-flavor
averaged branching fraction of BF(B --> pi l+l-) < 7.0 x 10^-8 and a
lepton-flavor averaged upper limit of BF(B0 --> eta l+l-) < 9.2 x 10^-8, both
at the 90% confidence level. We also report 90% confidence level branching
fraction upper limits for the individual modes B+ --> pi+ e+e-, B0 --> pi0
e+e-, B+ --> pi+ mu+mu-, B0 --> pi0 mu+mu-, B0 --> eta e+e-, and B0 --> eta
mu+mu-.
