  Let $(R,\fm)$ be a commutative Noetherian local ring. Suppose that $M$ and
$N$ are finitely generated modules over $R$ such that $M$ has finite projective
dimension and such that $\Tor^R_i(M,N)=0$ for all $i>0$. The main result of
this note gives a condition on $M$ which is necessary and sufficient for the
tensor product of $M$ and $N$ to be a Cohen--Macaulay module over $R$, provided
$N$ is itself a Cohen--Macaulay module.
