The diffraction of a plane electromagnetic wave by a parallel-plate waveguide cavity with a thick planar termination is rigorously analyzed for both the E and the H polarization using the Wiener-Hopf technique. Introducing the Fourier transform for the unknown scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations, which are solved exactly in a formal sense via the factorization and decomposition procedure. Since the formal solution involves an infinite number of unknowns and branch-cut integrals with unknown integrands, approximation procedures based on rigorous asymptotics are further presented to yield the approximate solution convenient for numerical computations. The scattered field inside and outside the cavity is evaluated by taking the inverse Fourier transform and applying the saddle point method. Representative numerical examples of the monostatic and bistatic radar cross sections are presented for various physical parameters, and the scattering characteristics of the cavity are discussed in detail.