Corollary

Let A be a local Krull domain and B a discrete valuation ring such that B dominates A and is aflat A-module. Then A is a field or a discrete valuation ring.

Let $\mathfrak{m}$ be the maximal ideal of B. By (PDE), $\mathfrak{m} \cap A$ is zero or of height 1. As it is, by hypothesis the maximal ideal of A, our assertion follows from Proposition 11 of no. 7.