The isometric immersion of two-dimensional Riemannian manifolds with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss--Codazzi equations for the first and second fundamental forms. The large solution is obtained, which leads to a isometric immersion. The approximate solutions are constructed by the Lax--Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The compactness is also derived. A compensated compactness framework is applied to obtain the existence of a large solution to the Gauss--Codazzi equations for surfaces that are more general than those in the literature.