The method of separation of variables is used to derive an analytic solution to the problem of electromagnetic scattering by a homogeneous dielectric spheroid with a confocal lossy dielectric coating of arbitrary thickness. The electric and magnetic fields in each different region are expressed in terms of a set of vector spheroidal eigenfunctions, and the solution is obtained by imposing the appropriate boundary conditions at each spheroidal surface. Computed results illustrate the dependence of the scattered field in the far zone on the coating material and its thickness, the size and the material of the spheroid, and on the angle of incidence.