This paper contains an exact solution for the stresses arising in an elastic circular cylinder with an oblate spheroidal cavity or an internal penny-shaped crack under tension. The solution is represented by a combination of the solution which is regular in the outside of a cavity and the solution which is regular in an infinite circular cylinder, and is deduced with the aid of Dougall's stress functions. Two sets of harmonic stress functions are given by simple expressions referred to the cylindrical and oblate spheroidal coordinates. The boundary conditions on the surfaces of the cylinder and of the cavity are satisfied by using the relations between the cylindrical and oblate spheroidal harmonics. And the solution for the internal penny-shaped crack is given for a limiting case of the solution for the oblate spheroidal cavity. Numerical results for stress distributions, stress concentration factors and stress intensity factors are shown graphically.