  We present evidence for D0-anti-D0 mixing in D0 --> K+pi- decays from 384
fb^{-1} of e+e- colliding-beam data recorded near sqrt(s) 10.6 GeV with the
BaBar detector at the PEP-II storage rings at SLAC. We find the mixing
parameters x'^2 = [-0.22 +- 0.30 (stat.) +- 0.21 (syst.)] x 10^{-3} and y' =
[9.7 +- 4.4 (stat.) +- 3.1 (syst.)] x 10^{-3}, and a correlation between them
of -0.94. This result is inconsistent with the no-mixing hypothesis with a
significance of 3.9 standard deviations. We measure R_D, the ratio of doubly
Cabibbo-suppressed to Cabibbo-favored decay rates, to be [0.303 +- 0.016
(stat.) +- 0.010 (syst.)]%. We find no evidence for \CP violation.
