ac.commutative algebra - Commutative Noetherian Domains of Krull Dimension One

This follows from a direct generalization of the Noether normalization lemma. It is covered in these notes from Mel Hochster. These notes prove it in a pretty general form (when the base ring is only an integral domain rather than a field).

Edit: A sufficient condition is that the algebra is finitely generated, but it is clearly not necessary.

Edit 2: I misread the question. I thought he was asking if A is finitely generated over some polynomial algebra (including infinitely generated polynomial algebras).