We describe a measurement of the direct CP asymmetry between inclusive b-->s gamma and b-->s gamma decays. This asymmetry is expected to be less than 0.01 in the standard model, but could be enhanced up to about 0.10 by new physics contributions. We use a sample of 89 x 10(6) BB pairs recorded with the BABAR detector at SLAC PEP-II, from which we reconstruct a set of 12 exclusive b-->s gamma final states containing one charged or neutral kaon and one to three pions. We measure an asymmetry of A(CP)(b-->s gamma)=0.025+/-0.050(stat)+/-0.015(syst), corresponding to an allowed range of -0.06<A(CP)(b-->s gamma)<+0.11 at 90% confidence level.