We consider a general class of birth-and-death processes with state space {0, 1, 2, 3,. . .} which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at 0 in the large population limit, that is, we scale the process by a parameter K, and take the limit K → +∞. We assume that the di erential equation dx/dt = b(x)−d(x) describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at 0, and an attractive fixed point x * > 0. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is n(b (x *) − d (x *)), n ≥ 0. The other one is the spectrum of a continuous-time binary branching process conditioned on nonextinction, and is given by n(d (0) − b (0)), n ≥ 1. A major di culty is that di erent scales and function spaces are involved. We work at the level of the eigenfunctions that we split over di erent regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min b (0) − d (0), d (x *) − b (x *). This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.