We report on a new measurement of the neutron β-asymmetry parameter A with the instrument Perkeo II. The main enhancements are the high neutron polarization of P = 99.7(1)% from a novel arrangement of supermirror polarizers and reduced background from improvements in beam line and shielding. The leading corrections were thus reduced by a factor of 4, pushing them below the level of statistical error and resulting in a significant reduction of systematic uncertainty compared to our previous experiments. We derive the β-asymmetry parameter A0 = -0.11972(45)(stat)((-44)(+32))(sys) = -0.11972((-65)(+53)) and the ratio of the axial vector to the vector coupling constant λ = gA/gV = -1.2761(12)(stat)((-12)(+9))(sys) = -1.2761((-17)(+14)).