integral closure is intersection of discrete valuation rings? In the book Introduction to Commutative algebra (Atiyah and Macdonald), Corollary 5.22 reads:
Let $A\subset K$ be a subring of a field $K$, then the integral closure of $A$ is intersection of all valuation rings of $K$ containing $A$.
Is it still true if we replace "valuation rings" above by "discrete valuation rings"? 
 A: Without some other major assumptions on $A$, this is false.  An important class of domains for which it is does hold are the Krull domains, which includes UFDs.  As pointed out in the comments it is also true for Noetherian domains, which can be argued directly or taken for granted from e.g. the Mori-Nagata theorem and the statement for Krull domains.
The easiest counterexample comes from valuation domains, because a valuation domain is integrally closed and it is the intersection of discrete valuation rings precisely when it is a DVR itself. (An easy way to see this is to note that overrings of a valuation domain are just localizations at prime ideals, and that a DVR is 1-dimensional.)
If you are familiar with the concept of complete integral closure, then note also that your hypothesis would imply that the complete integral closure of $A$ coincides with the integral closure of $A$, which is usually not the case, even with significant chain conditions like having ACCP (or even stronger being Mori).
