Proposition 18
Tag 04O7, permanent. Cite it as [Bourbaki, Tag 04O7].
Commutative Algebra, chapter V, § 1. Notion of an integral element
Let A be an integrally closed domain, K its field of fractions, K' a finite-dimensional separable K-algebra (Algebra, Chapter VIII, § 7, no. 5) and A' the integral closure of A in K'. Then A' is contained in afinitely generated A-module.
The proposition will follow from the following more precise lemma:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.