The factorization of Wiener–Hopf matrices with exponentially growing elements has long remained an unsolved problem. Such matrices often occur in the calculation of scattering from complex canonical geometries [Abrahams and Wickham, Proc. Roy. Soc. London, Ser. A, pp. 131–156] and other problems in mathematical physics. Here a general method is given for resolving the difficulty and obtaining a single scalar integral equation, the solution of which generates the required factors. It is proven that this solution has a simple infinite series representation. The method is illustrated by reference to three particular examples. The first is a model matrix arising from a system of simple delay–differential equations and the second occurs in the diffraction of sound by two “knife-edges.” Finally a heat conduction problem is solved for an infinite wedge with asymmetric mixed boundary conditions on its faces.