The diffraction of a plane electromagnetic wave by a perfectly-conducting finite sinusoidal grating is analyzed using the Wiener-Hopf technique combined with the perturbation method. Assuming the depth of the grating to be small compared with the wavelength and approximating the boundary condition on the grating surface, the problem is reduced to that of the diffraction by a flat strip with a certain mixed boundary condition. Introducing a perturbation expansion for the unknown scattered field under the small-depth approximation, the problem is formulated in terms of the zero order and the first order Wiener-Hopf equations, which are solved exactly in a formal sense by a factorization and decomposition procedure. Applying a rigorous asymptotics, explicit high-frequency asymptotic solutions are further obtained for the width of the grating large compared with the wavelength. Scattered far field expressions are derived by taking the inverse Fourier transform are applying the saddle point method. Representative numerical examples of the far field patterns are given for various physical parameters, and the scattering characteristics of the grating are discussed.