  Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose
that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let
the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections
$C_1, \ldots ,C_m$. We show that there is a reduced local ring $S \subseteq B$
with maximal ideal $S \cap M$ such that the completion of $S$ with respect to
its maximal ideal is isomorphic to the completion of $B$ with respect to its
maximal ideal and such that, if $P$ and $Q$ are prime ideals of $B$, then $P
\cap S = Q \cap S$ if and only if $P$ and $Q$ are in $C_i$ for some $i = 1,2,
\ldots ,m$.
