Corollary 2
Tag 04ZB, permanent. Cite it as [Bourbaki, Tag 04ZB].
Commutative Algebra, chapter VII, § 1. Krull domains
Let A be a Krull domain, K its field of fractions, v a valuation on K which is positive on A and p the set of x ∈ A such that v(x) > 0. If the prime ideal p is of height 1, v is equivalent to an essential valuation of A.
Let B be the ring of v and m its ideal. Then m ∩ A = p and hence A_p ⊂ B. Now A_p is a discrete valuation ring (Theorem 3 and Corollary 1 to Proposition 6). As p ≠ (0), B ≠ K and hence B = A, (Chapter VI, § 4, no. 5, Proposition 6).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.