We measure time-integrated CP-violating asymmetries of neutral charmed mesons in the modes D0 --> K(-) K(+) and D0 --> pi(-) pi(+) with the highest precision to date by using D0 --> K(-) pi(+) decays to correct detector asymmetries. An analysis of 385.8 fb(-1) of data collected with the BABAR detector yields values of a(CP)(KK)=(0.00+/-0.34(stat)+/-0.13(syst))% and a(CP)(pipi)=(-0.24+/-0.52(stat)+/-0.22(syst))%, which agree with standard model predictions.