Let k be a subring of the field of rational functions in , s which contains and s . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of k={ Q}( , s), the field of rational functions in , s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S S ) is freely generated by the empty link, i.e., K(S S )