We report measurements of the branching fractions of neutral and charged B meson decays to final states containing a K_1(1270) or K_1(1400) meson and a charged pion. The data, collected with the BABAR detector at the SLAC National Accelerator Laboratory, correspond to 454×10^6 BB̅ pairs produced in e^+e^- annihilation. We measure the branching fractions B(B^0→K_1(1270)^+π-+K_1(1400)^+π-)=3.1_(-0.7)^(+0.8)×10^(-5) and B(B^+→K_1(1270)^0π^++K_1(1400)^0π^+)=2.9_(-1.7)^(+2.9)×10^(-5) (<8.2×10^(-5) at 90% confidence level), where the errors are statistical and systematic combined. The B^0 decay mode is observed with a significance of 7.5σ, while a significance of 3.2σ is obtained for the B^+ decay mode. Based on these results, we estimate the weak phase α=(79±7±11)° from the time-dependent CP asymmetries in B^0→a1(1260)^±π^∓ decays
