Summary: An asymptotic solution is obtained for the problem of the diffraction of a whispering-gallery mode by a spherical cap. The convex side of the cap has a soft (pressure release) boundary condition and the concave side a rigid boundary condition. By using an integral transform with kernel \(Q^ 0_{\nu -1/2}(\cos \theta)\), the mixed boundary-value problem is reduced into a matrix Hilbert equation which is solved by following a method initially introduced by R. A. Hurd [The Wiener-Hopf-Hilbert method for diffraction problems. Canad. J. Phys. 54, 775-780 (1976)] and extended by A. D. Rawling and W. E. Williams [Q. J. Mech. Appl. Math. 34, 1-8 (1981; Zbl 0458.15010)]. Various ray contributions are isolated and fairly simple non-uniform expressions for certain diffraction coefficients are obtained.