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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 withthe full EXO-200 dataset. A deep learning-based convolutional neural network isused to discriminate signal from background events. Signal detection efficiencyis increased relative to previous searches by EXO-200 by more than a factor oftwo. 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 statisticallysignificant evidence for $2\nu\beta\beta$ decay to the $0^+_1$ state isobserved, 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.

Search for Two-neutrino Double-Beta Decay of $^{136}\rm Xe$ to the $0^+_1$ excited state of $^{136}\rm Ba$ with the Complete EXO-200 Dataset