This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example.

To state the question again: let $A$ be an integral Noetherien ring of Krull dimension one and $K$ its field of fractions. Let $B$ be the set of elements of $K$ that are integral over $A$ i.e. $B$ is the normalisation of $A$ in $K$.

Is the morphism $A \to B$ finite?

Note that this is true if $A$ is excellent (or even just Nagata), and its rather difficult to construct examples of non-excellent rings.