  This article presents the measurement of a $t$-channel single top-quark
production fiducial cross-section in the lepton+jets channel with 20.3
fb$^{-1}$ of 8 TeV data using a neural-network discriminant. A fiducial
cross-section quoted within the detector acceptance of $\sigma_{\rm fid} = 3.37
\pm 0.05 \, (\mathrm{stat.}) \pm 0.47 \, (\mathrm{syst.}) \pm 0.09 \,
(\mathrm{lumi.})~\text{pb}$ is obtained. The total inclusive $t$-channel
cross-section is calculated using the acceptance predicted by various Monte
Carlo generators. If the acceptance from the a MC@NLO + Herwig event generator
is used, a value of $\sigma_t = 82.6 \pm 1.2 \, (\mathrm{stat.}) \pm 11.4 \,
(\mathrm{syst.}) \pm 3.1 \, (\mathrm{PDF}) \pm 2.3 \,
(\mathrm{lumi.})~\text{pb}$ is obtained, consistent with the Standard Model
prediction. Using the ratio of the measured inclusive cross-section to the
predicted cross-section and assuming that the top-quark-related CKM matrix
elements obey the relation $|V_{tb}|\gg |V_{ts}|, |V_{td}|$, the coupling
strength at the $W$-$t$-$b$ vertex is determined to be
$|V_{tb}|=0.97^{+0.09}_{-0.10}$. Assuming that $|V_{tb}|\leq 1$ a lower limit
of $|V_{tb}|>0.78$ is obtained at the 95% confidence level.
