Double-beta decay is a rare nuclear process in which two neutrons in the nucleus are converted to two protons with the emission of two electrons and two electron anti-neutrinos. We measured the half life of the two-neutrino double-beta decay of $^{150}$Nd to excited final states of $^{150}$Sm by detecting the de-excitation gamma rays of the daughter nucleus. This study yields the first detection of the coincidence gamma rays from the 0$^+_1$ excited state of $^{150}$Sm. These gamma rays have energies of 333.97 keV and 406.52 keV, and are emitted in coincidence through a 0$^+_1rightarrow$2$^+_1rightarrow$0$^+_{gs}$ transition. The enriched Nd$_2$O$_3$ sample consisted of 40.13 g $^{150}$Nd and was observed for 642.8 days at the Kimballton Underground Research Facility, producing 21.6 net events in the region of interest. This count rate gives a half life of $T_{1/2}=(1.07^{+0.45}_{-0.25}(stat)pm0.07(syst.))times 10^{20}$ years. The effective nuclear matrix element was found to be 0.0465$^{+0.0098}_{-0.0054}$. Finally, lower limits were obtained for decays to higher excited final states. Our half-life measurement agrees within uncertainties with another recent measurement in which no coincidence was employed. Our nuclear matrix element calculation may have an impact on a recent neutrinoless double-beta decay nuclear matrix element calculation which implies the decay to the first excited state in $^{150}$Sm is favored over that to the ground state.
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