  The neutron $\beta$-decay asymmetry parameter $A_0$ defines the correlation
between the spin of the neutron and the momentum of the emitted electron, which
determines $\lambda=\frac{g_{A}}{g_{V}}$, the ratio of the axial-vector to
vector weak coupling constants. The UCNA Experiment, located at the Ultracold
Neutron facility at the Los Alamos Neutron Science Center, is the first to
measure such a correlation coefficient using ultracold neutrons (UCN).
Following improvements to the systematic uncertainties and increased
statistics, we report the new result $A_0 =
-0.12054(44)_{\mathrm{stat}}(68)_{\mathrm{syst}}$ which yields $\lambda\equiv
\frac{g_{A}}{g_{V}}=-1.2783(22)$. Combination with the previous UCNA result and
accounting for correlated systematic uncertainties produces
$A_0=-0.12015(34)_{\mathrm{stat}}(63)_{\mathrm{syst}}$ and $\lambda\equiv
\frac{g_{A}}{g_{V}}=-1.2772(20)$.
