An approximate asymptotic high-frequency solution is obtained for the electromagnetic wave diffraction by a wedge with convex faces and a moderate wedge angle. The present solution properly includes the edge-excited surface ray effect and can predict the transition between the space ray and the surface ray. This solution is uniform across the various ray shadow boundaries even when the transition regions associated with these ray shadow boundaries overlap and is expressed in the simple format of the geometrical theory of diffraction. The ansatz for the present solution is based on a uniform solution for the scattering from a curved screen obtained earlier by C.W. Chuang and M.C. Liang, used in conjunction with an asymptotic matching technique. Numerical results based on the present uniform geometrical theory of diffraction solution are found to be in excellent agreement with those based on an independent moment method solution of the exact integral equation.