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EXO-200 is a single phase liquid xenon detector designed to search forneutrinoless 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 $^{136}$Xeto the first $0^+$ excited state, $0^+_1$, of $^{136}$Ba based on a 100kg$\cdot$yr exposure of $^{136}$Xe. Using a specialized analysis employing amachine 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$2\nu\beta\beta$ decay to the excited state resulting in a lower limit of$T^{2\nu}_{1/2}$ ($0^+ \rightarrow 0^+_1$) $> 6.9 \times 10^{23}$ yr at 90% CL.This observed limit is consistent with the estimated half-life of$2.5\times10^{25}$ yr.

Search for $2\nu\beta\beta$ decay of $^{136}$Xe to the 0$_1^+$ excited state of $^{136}$Ba with EXO-200