On the rigidity of small domains

Let k be an algebraically closed field of arbitrary characteristic. Let A be an affine domain over k with transcendence degree 1 which is not isomorphic to k[x], and let B be a domain over k. We show that the AK invariant distributes over the tensor product of A by B. As a consequence, we obtain a generalization of the cancellation theorem of S. Abhyankar, P. Eakin, and W. Heinzer.