The boundary integral equation method is used to solve the problem corresponding to antiplane shear deformation of a homogeneous and anisotropic linearly elastic solid whose cross-section is bounded by an arbitrary (smooth) closed curve. The solution is found in the form of a single-layer potential based on the principal fundamental solution of antiplane shear. Uniqueness and existence results are established in the appropriate function spaces. An example of an elastic solid with an elliptic cross-section is used to illustrate the theory.