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 B ¯¯¯ B pairs produced in e + e − annihilation. We measure the branching fractions B ( B 0 → K 1 ( 1270 ) + π − + K 1 ( 1400 ) + π − ) = 3.1 + 0.8 − 0.7 × 10 − 5 and B ( B + → K 1 ( 1270 ) 0 π + + K 1 ( 1400 ) 0 π + ) = 2.9 + 2.9 − 1.7 × 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 7.5 σ , while a significance of 3.2 σ is obtained for the B + decay mode. Based on these results, we estimate the weak phase α = ( 79 ± 7 ± 11 ) ° from the time-dependent C P asymmetries in B 0 → a 1 ( 1260 ) ± π ∓ decays.