Suppose $f: A \to B$ is a local homomorphism, $B$ is isomorphic to a localization of an $A$-algebra of finite type. Let $L$ be the field of fractions of $B$, and suppose that $B$ contains the normal closure $A'$ of $A$ in $L$.
Grothendieck writes (SGA 1, second proof of Theorem 9.5., page 16) that Zariski's Main Theorem implies that $B$ is isomorphic to a localization of $A'$.
Why is this so? I tried applying some algebraic versions of ZMT, but failed to see the implication.