  A search for neutrinoless double-beta decay ($0\nu\beta\beta$) in $^{136}$Xe
is performed with the full EXO-200 dataset using a deep neural network to
discriminate between $0\nu\beta\beta$ and background events. Relative to
previous analyses, the signal detection efficiency has been raised from 80.8%
to 96.4$\pm$3.0% and the energy resolution of the detector at the Q-value of
$^{136}$Xe $0\nu\beta\beta$ has been improved from $\sigma/E=1.23\%$ to
$1.15\pm0.02\%$ with the upgraded detector. Accounting for the new data, the
median 90% confidence level $0\nu\beta\beta$ half-life sensitivity for this
analysis is $5.0 \cdot 10^{25}$ yr with a total $^{136}$Xe exposure of 234.1
kg$\cdot$yr. No statistically significant evidence for $0\nu\beta\beta$ is
observed, leading to a lower limit on the $0\nu\beta\beta$ half-life of
$3.5\cdot10^{25}$ yr at the 90% confidence level.
