We construct a numerical algorithm for the approximate solution of a nonlinear elastic limiting strain model based on the Fourier spectral method. The existence and uniqueness of the numerical solution are proved. Assuming that the weak solution to the boundary-value problem possesses suitable Sobolev regularity, the sequence of numerical solutions is shown to converge to the weak solution of the problem at an optimal rate. The numerical method represents a finite-dimensional system of nonlinear equations. An iterative method is proposed for the approximate solution of this system of equations and is shown to converge, at a linear rate, to the unique solution of the numerical method. The theoretical results are illustrated by numerical experiments.