The Jacobson radical J(K[S]) of the semigroup ring K[S] of a cancellative semigroup S over a field K is studied. We show that, if J(K[S]) ≠ 0, then either S is a reversive semigroup or K[S] has many nilpotents and J(K[P]) ≠ 0 for a reversive subsemigroup P of S. This is used to prove that J(K[S]) = 0 for every unique product. semigroup S.