  We present a detailed report of a measurement of the neutron
$\beta$-asymmetry parameter $A_0$, 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 $2 \times 2\pi$ 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. From an analysis of data
acquired during runs in 2008 and 2009, we report $A_0 = -0.11966 \pm
0.00089_{-0.00140} ^{+0.00123}$, from which we extract a value for the ratio of
the weak axial-vector and vector coupling constants of the nucleon, $\lambda =
g_A/g_V = -1.27590 \pm 0.00239_{-0.00377}^{+0.00331}$. Complete details of the
analysis are presented.
