Corollary 3

Let $A$ be an integral domain and $\Omega$ the set of maximal ideals of $A$. For $A$ to be integrally closed, it is necessary and sufficient that, for all $m \in \Omega$, $A_m$ be integrally closed.

It follows from Corollary 1 that the condition is necessary. The condition is sufficient, for $A = \bigcap_{m \in \Omega} A_m$ (Chapter II, § 3, no. 3, formula (2)) and it is sufficient to apply the Corollary to Proposition 8 of no. 2.