In this paper, the axisymmetric torsion problem of a compound infinite cylinder with an external annular crack is considered on the basis of the three-dimensional theory of elasticity. This problem is reduced to the solution of an infinite system of simultaneous equations. The coefficient matrix is given by the product of three matrices including an inverse matrix. Numerical results are illustrated for the distributions of displacement and shear stress and for the variations of the stress intensity factor K_III for various ratios of inner to outer material shear modulus and various ratios of inner to outer radius of the cylinder.