  A new search for two-neutrino double-beta ($2\nu\beta\beta$) decay of
$^{136}\rm Xe$ to the $0^+_1$ excited state of $^{136}\rm Ba$ is performed with
the full EXO-200 dataset. A deep learning-based convolutional neural network is
used to discriminate signal from background events. Signal detection efficiency
is increased relative to previous searches by EXO-200 by more than a factor of
two. With the addition of the Phase II dataset taken with an upgraded detector,
the median 90$\%$ confidence level half-life sensitivity of $2\nu\beta\beta$
decay to the $0^+_1$ state of $^{136}\rm Ba$ is $2.9 \times 10^{24}~\rm yr$
using a total $^{136}\rm Xe$ exposure of $234.1~\rm kg~yr$. No statistically
significant evidence for $2\nu\beta\beta$ decay to the $0^+_1$ state is
observed, leading to a lower limit of $T^{2\nu}_{1/2}(0^+ \rightarrow 0^+_1) >
1.4\times10^{24}~\rm yr$ at 90$\%$ confidence level, improved by 70$\%$
relative to the current world's best constraint.
