In this study, the mode III axisymmetric crack problem in a non-homogeneous interfacial region between two homogeneous half-spaces is considered. The shear modulus of the interfacial layer is assumed to be mu(2)(z) = mu(0)e(mz). It is also assumed that this shear modulus varies continuously between that of the two half-spaces. By using the Hankel transform technique the problem is reduced to a singular integral equation. The problem is solved for various material combinations, crack geometries and for three different sets of crack surface tractions, and the corresponding stress intensity factors are tabulated.