EXO-200 is a single phase liquid xenon detector designed to search for neutrinoless double-beta decay of $^{136}$Xe to the ground state of $^{136}$Ba. We report here on a search for the two-neutrino double-beta decay of $^{136}$Xe to the first $0^+$ excited state, $0^+_1$, of $^{136}$Ba based on a 100 kg$cdot$yr exposure of $^{136}$Xe. Using a specialized analysis employing a machine learning algorithm, we obtain a 90% CL half-life sensitivity of $1.7 times 10^{24}$ yr. We find no statistically significant evidence for the $2 ubetabeta$ decay to the excited state resulting in a lower limit of $T^{2 u}_{1/2}$ ($0^+ rightarrow 0^+_1$) $> 6.9 times 10^{23}$ yr at 90% CL. This observed limit is consistent with the estimated half-life of $2.5times10^{25}$ yr.
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