Let R be a hypersurface domain, that is, a ring of the form S/(f), where S is a regular local ring and f is a prime element. Suppose M and N are finitely generated R- modules. We prove two rigidity theorems on the vanishing of Tor. In the first theorem we assume that the regular local ring S is unramified, that M R N has finite length, and that dim(M)+dim(N) dim(R). With these assumptions, if Tor R (M,N) = 0 for some j 0, then Tor R (M,N) = 0 for all i j. The second rigidity theorem states that if M R N is reflexive, then Tor R (M,N) = 0 for all i 1. We use these theorems to prove the following theorem (valid even if S is ramified): If M R N is a maximal Cohen-Macaulay R-module, then both M and N are maximal Cohen-Macaulay modules, and at least one of them is free.