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 $2 times 2pi$ 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.
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