We measure the time dependence of the ratio of decay rates for D0 -> K+ pi- to the Cabibbo-favored decay D0 -> K- pi+. The charge conjugate decays are included. A signal of 3.3 x 10^4 D*+ -> pi+ D0, D0 -> K+ pi- decays is obtained with D0 proper decay times between 0.75 and 10 mean D0 lifetimes. The data were recorded with the CDF II detector at the Fermilab Tevatron and correspond to an integrated luminosity of 9.6 fb-1 for p-pbar collisions at sqrt(s) = 1.96 TeV. Assuming CP conservation, we search for D0-D0bar mixing and measure the mixing parameters to be R_D = (3.51 +/- 0.35) x 10^{-3}, y = (4.3 +/- 4.3) x 10^{-3}, and x^2 = (0.08 +/- 0.18) x 10^{-3}. We report Bayesian probability intervals in the x^2 - y plane and find that the significance of excluding the no-mixing hypothesis is equivalent to 6.1 Gaussian standard deviations, providing the second observation of D0-D0bar mixing from a single experiment.
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