(J, J')-lossless factorization plays a central role in H-infinity-control because it gives a simple and unified framework of H-infinity-control from the viewpoint of classical network theory, and it includes the well-known inner-outer factorization of rational matrices, Wiener-Hopf factorization, and spectral factorization of positive rational matrices as special cases. However, up to now, there is still a lack of numerically reliable methods for this important factorization problem in a general setting. In this paper, we present necessary and sufficient solvability conditions and develop a numerically reliable algorithm based on a generalized eigenvalue approach for the (J, J')-lossless factorization of general rational matrices.