We present a detailed report of a measurement of the neutron β-asymmetry parameter A0, the parity-violating angular correlation between the neutron spin and the decay electron momentum, performed with polarized ultracold neutrons (UCN). UCN were extracted from a pulsed spallation solid deuterium source and polarized via transport through a 7-T magnetic field. The polarized UCN were then transported through an adiabatic-fast-passage spin-flipper field region, prior to storage in a cylindrical decay volume situated within a 1-T 2 × 2π solenoidal spectrometer. The asymmetry was extracted from measurements of the decay electrons in multiwire proportional chamber and plastic scintillator detector packages located on both ends of the spectrometer.