The purpose of this paper is to prove a uniform extendibility result for sequences of biholomorphic mappings between weakly pseudoconvex domains. Suppose that f21 and 02 are bounded weakly pseudoconvex domains in {E" with real analytic boundaries and suppose that {fi} is a sequence of biholomorphic mappings J~ : I2t--,O2 which converges to a holomorphic mapping f of ~2~ into the closure of 122. (We remark that a strictly pseudoconvex domain is not a weakly pseudoconvex domain.) Cartan's Theorem states that either f is a biholomorphic map of f2~ onto 02, or f is a mapping of f21 into the boundary of f22. We shall prove that, in either case, there exists a domain D which contains 121 together with all the strictly pseudoconvex boundary points of f2x such that each of the mappings f~ extend to be hoIomorphic on D. Furthermore, the limit map fextends holomorphically to D and the sequence {fi} converges uniformly on compact subsets of D to f. It is interesting to note that the corresponding statement for a sequence of maps between strictly pseudoconvex domains with real analytic boundaries is not true. Indeed, if I21 is strictly pseudoconvex and the limit map f maps into bf22, then both O t and I22 are biholomorphic to the ball by Wong's theorem. In this situation, if the inverse mappings Fi =f~-1 also converge, then there is an open set D which contains the closure of O1 minus a single point in bf21 to which all the f~ extend, and the sequence f~ converges uniformly on compact subsets of D to f. If no assumption is made about the convergence of the inverses, then the maps./~ need not converge uniformly up to the boundary near any boundary point of O1. When f21 and 02 are assumed to be weakly pseudoconvex, and therefore not biholomorphic to the ball, the assumption about the convergence of the inverse maps is not needed.