The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to the conclusion)?

Theorem. Let $R$ be a commutative UFD with field of fractions $F$. Suppose that for any subring $S$ of $F$ that properly includes $R$, there is some non-unit of $R$ that becomes invertible in $S$. Then $R$ is a PID.