  We measure 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 aCP(KK) = (0.00 +/-
0.34 (stat.) +/- 0.13 (syst.))% and aCP(pipi) = (-0.24 +/- 0.52 (stat.) +/-
0.22 (syst.))%, which agree with Standard Model predictions.
