Apologies in advance if this is too elementary.

The following is well known when $A$ is an algebraically closed field:

Let $X$ be an integral closed subscheme of $P^n_A$. Then $\Gamma(X, \mathcal{O}_X) = A$.

My question is: For what other rings does the above statement hold?

There are two proofs of this (for $A$ algebraically closed) in Hartshorne. Both seem to use the fact that the integral closure of $A$ in its quotient field is just $A$ itself in a key way.

So I suspect that having $A$ integrally closed will be crucial, but I do not know. In particular, does the proof in Hartshorne still work, and if so, does it apply to nonnormal domains?