In this paper, we are interested in the numerical approximation of the method for computing guided modes in integrated optics introduced in our earlier paper [Gómez Pedreira and Joly, SIAM J. Numer. Anal., 39 (2001), pp. 596--623]. We saw that the main step of the computation of the guided modes consists of computing the eigenvalues of a compact operator ${\rm K}$ depending on the frequency $\omega$ and the wavenumber $\beta$. This second paper is devoted to the discretization of this problem and to the corresponding convergence analysis. In particular, we derive error estimates in the eigenvalue approximation, which will be of exponential accuracy. Copyright © 2002 Society for Industrial and Applied Mathematics