This question is motivated by this.
Let $k$ be a field. Let $K$ be a finitely generated extension field of $k$ of transcendence degree one. Let $A$ be a subring of $K$ containing $k$. Suppose that $A$ contains a transcendental element over $k$. Let $B$ be the integral closure of $A$ in $K$. Is $B$ finitely generated as an $A$-module?