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We present a measurement of relative partial widths and decay rate CP asymmetries in K-K+ and π- π+ decays of D0 mesons produced in pp̅ collisions at √s=1.96 TeV. We use a sample of 2×105 D*+ D0 π+ (and charge conjugate) decays with the D0 decaying to K- π +, K-K+, and π- π+, corresponding to 123 pb-1 of data collected by the Collider Detector at Fermilab II experiment at the Fermilab Tevatron collider. No significant direct CP violation is observed. We measure Γ (D0→ K-K+)/ Γ (D0→ K- π +)=0.0992±0.0011±0.0012, Γ (D0→ π- π+)/ Γ (D0→ K- π +)=0.035 94±0.000 54±0.000 40, ACP(K-K+)=(2.0±1.2±0.6)%, and ACP(π- π+)=(1.0±1.3±0.6)%, where, in all cases, the first uncertainty is statistical and the second is systematic.