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 during 5.25 y with 36.6 g of the isotope $^{150}$Nd were used in the analysis. For the first time the signal of $2\nu\beta\beta$ transition to the 0$^+_1$ excited state is detected with statistical significance exceeding 5 sigma. The half-life is measured to be $T_{1/2}^{2\nu\beta\beta}(0^+_1) = \left[ 1.11 ^{+0.19}_{-0.14} \,\left(\mbox{stat}\right) ^{+0.17}_{-0.15}\, \left(\mbox{syst}\right) \right] \times10^{20}\,\mbox{y}$. Limits are set on $2\nu\beta\beta$ decay to 2$^+_1$ level and on $0\nu\beta\beta$ decay to 0$^+_1$ and 2$^+_1$ levels of $^{150}$Sm.
