<p>We introduce a numerical method which is developed to approach the solution to a nonlinear age-structured population model with a finite life-span. The new scheme employs given numerical approximations to the intrinsic survival probability to set a full discretization of the problem, in which the numerical approach to the age-specific density of the population is separated from the difficulties arising due to the singular mortality rate. Its convergence is completely established without an explicit dependence on the asymptotic behaviour of the natural mortality rate near the maximum age. We include a numerical experimentation that confirms numerically the second-order convergence of the approximations provided by the corresponding theorems.</p>