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.