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The NEMO-3 results for the double-$\beta$ decay of $^{150}$Nd to the 0$^+_1$and 2$^+_1$ excited states of $^{150}$Sm are reported. The data recorded during5.25 yr with 36.6 g of the isotope $^{150}$Nd are used in the analysis. For thefirst time, the signal of the $2\nu\beta\beta$ transition to the 0$^+_1$excited state is detected with a statistical significance exceeding 5$\sigma$.The half-life is measured to be $T_{1/2}^{2\nu\beta\beta}(0^+_1) = \left[ 1.11^{+0.19}_{-0.14} \,\left(\mbox{stat}\right) ^{+0.17}_{-0.15}\,\left(\mbox{syst}\right) \right] \times10^{20}\,\mbox{yr}$. The limits are seton the $2\nu\beta\beta$ decay to the 2$^+_1$ level and on the $0\nu\beta\beta$decay to the 0$^+_1$ and 2$^+_1$ levels of $^{150}$Sm.

Measurement of double-$\beta$ decay of $^{150}$Nd to the 0$^+_1$ excited state of $^{150}$Sm in NEMO-3