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We present improved measurements of CP-violation parameters in the decays$B^0 \to \pi^+ \pi^-$, $B^0 \to K^+ \pi^-$, and $B^0 \to \pi^0 \pi^0$, and ofthe branching fractions for $B^0 \to \pi^0 \pi^0$ and $B^0 \to K^0 \pi^0$. Theresults are obtained with the full data set collected at the $\Upsilon(4S)$resonance by the BABAR experiment at the PEP-II asymmetric-energy $B$ factoryat the SLAC National Accelerator Laboratory, corresponding to $467 \pm 5$million $B\bar B$ pairs. We find the CP-violation parameter values andbranching fractions $ S_{\pi^+\pi^-} = -0.68 \pm 0.10 \pm 0.03, C_{\pi^+\pi^-}= -0.25 \pm 0.08 \pm 0.02, A_{K^-\pi^+} = -0.107 \pm 0.016 ^{+0.006}_{-0.004},C_{\pi^0\pi^0} = -0.43 \pm 0.26 \pm 0.05, Br(B^0 \to \pi^0 \pi^0) = (1.83 \pm0.21 \pm 0.13) \times 10^{-6}, Br(B^0 \to K^0 \pi^0) = (10.1 \pm 0.6 \pm 0.4)\times 10^{-6},$ where in each case, the first uncertainties are statisticaland the second are systematic. We observe CP violation with a significance of6.7 standard deviations for $B^0 \to\pi^+\pi^-$ and 6.1 standard deviations for$B^0 \to K^+ \pi^-$, including systematic uncertainties. Constraints on theUnitarity Triangle angle $\alpha$ are determined from the isospin relationsamong the $B \to \pi\pi$ rates and asymmetries. Considering only the solutionpreferred by the Standard Model, we find $\alpha$ to be in the range$[71^\circ,109^\circ]$ at the 68% confidence level.

Measurement of CP Asymmetries and Branching Fractions in Charmless Two-Body B-Meson Decays to Pions and Kaons