  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$^+_1\rightarrow$2$^+_1\rightarrow$0$^+_{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.
