Let \((R,M)\) be an excellent normal local Henselian domain with completion \(\widehat R\) and field of fractions \({\mathcal Q} (R)\). It is shown that, if \(Q\) is a prime ideal of height \(\geq 2\) in \(R\) such that \(R/Q\) is not complete, then there are infinitely many height one prime ideals \(P \subseteq Q \widehat R\) of \(\widehat R\) with \(P \cap R=0\); in particular, the dimension of the generic formal fiber of \(R\) is at least one. The authors consider connections with the question as to whether excellent Henselian local rings \(R\) have the following property: Whenever \(P\) is a prime ideal of \(\widehat R\) which is maximal in the generic formal fiber of \(R\) it follows that, for all but at most finitely many of the prime ideals \(Q \supseteq P\) in \(\widehat R\), the ring \(R/(Q \cap R)\) is complete. Various examples are discussed in connection with this property.
The second half of the paper concerns excellent normal local Henselian domains \(R\) with zero-dimensional formal fibers. Such an \(R\) has the property that, for any field \(L\) between \({\mathcal Q} (R)\) and \({\mathcal Q} (\widehat R)\), the ring \(L \cap \widehat R\) is a local Noetherian domain which has completion \(\widehat R\).