Let $A$ be a one-dimension local noetherian domain and suppose that we know that $K=\text{Frac}(A)$ is a complete discrete valuation field (valuations for me are surjective). Let's denote with $\mathcal O_K$ the valuation ring of $K$.
In general $\mathcal O_K\neq A$ and I suppose that $\mathcal O_K\supseteq A$ . Is $\mathcal O_K$ the integral closure of $A$ in $K$?
The integral closure of a noetherian domain is the intersection of all the DVRs of its field of fractions, so the answer to your question is yes.