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 \(2\nu\beta\beta\) 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.