  Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be
the space of valuation rings of $F$ that dominate $D$. We lift Zariski's
connectedness theorem for fibers of a projective morphism to the
Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$
of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally
closed in $F$ if and only if there is a closed and connected subspace $Z$ of
Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently,
the intersection of the rings in any closed and connected subset of Val$_F(D)$
is a local ring. In proving this, we also prove a converse to Zariski's
connectedness theorem. Our results do not require the rings involved to be
Noetherian.
