Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?

It is a standard fact that Krull domains, and thus noetherian normal rings, have this property. But Krull domains satisfy two additional properties, namely:

- $R_P$ is a discrete valuation ring for all height one primes $P$.
- Every non-zero element is contained in only a finite number of height one prime ideals.

What happens if I drop these two assumptions? Do I get anything new? Is this true for non-noetherian normal rings (I fear not)?