Prime ideals in birational extensions of two-dimensional polynomial rings

“For which commutative Noetherian rings A is Spec(A), the set of prime ideals of A, order-isomorphic under inclusion to image, the prime ideals of the polynomial ring in one variable over the integers?” We show that this is true for every finitely generated birational extension of the polynomial ring in one variable over an order D in an algebraic number field; that is, if B is an intermediate ring between D[x] and its quotient field and B is finitely generated over D[x], then image.