  Let $A$ be a local Noetherian domain of Krull dimension $d$. Heinzer,
Rotthaus and Sally have shown that if the generic formal fiber of $A$ has
dimension $d-1$, then $A$ is birationally dominated by a one-dimensional
analytically ramified local Noetherian ring having residue field finite over
the residue field of $A$. We explore further this correspondence between prime
ideals in the generic formal fiber and one-dimensional analytically ramified
local rings. Our main focus is on the case where the analytically ramified
local rings are stable, and we show that in this case the embedding dimension
of the stable ring reflects the embedding dimension of a prime ideal maximal in
the generic formal fiber, thus providing a measure of how far the generic
formal fiber deviates from regularity. A number of characterizations of
analytically ramified local stable domains are also given.
