The diffraction of a Kelvin wave by a transverse (to a straight coastline) discontinuity in depth is considered. A Fourier-integral formulation leads to a singular integral equation that may be solved exactly; however, the integrals in this solution are intractable without further approximation. An expansion to third order in a depth-change parameter yields results that are generally adequate for tidal problems (such as that posed by the Mendocino fracture zone) but are inadequate for the double-Kelvin-wave regime. Approximations are developed for a continuous change of depth that is either small or gradual, and the diffracted Kelvin wave along the coastline is found to have an amplitude that is inversely proportional to the square root of the depth and a phase that is given by the integral of the wavenumber.