Birationality of affine line implies birationality?

Question

Let $A,B$ be noetherian regular domains of dimension one, such that $\mathbb{A}^1_A$ is birational to $\mathbb{A}^1_B$. Is $\text{Spec}(A)$ birational to $\text{Spec}(B)$?

Answer 1

In many cases, the answer is yes:

Theorem [Deveney 1982, Theorem 2]. Let $k$ be a field, and let $L$ be a finitely generated field extension of $k$. For $i = 1,2$, consider an intermediate subfield $L \supseteq L_i \supseteq k$ finitely generated over $k$, such that $\operatorname{trdeg}_k L_i = 1$. If $L = L_1(x_1) = L_2(x_2)$ for some elements $x_1,x_2 \in L$ transcendental over $L_1,L_2$ respectively, then $L_1 \simeq L_2$.

You can apply this theorem to your situation when $A$ and $B$ are one-dimensional domains essentially of finite type over a field $k$, in which case you set $L_1 = \operatorname{Frac} A$ and $L_2 = \operatorname{Frac} B$.

I don't know the answer when you are not working over a field.

References

[Deveney 1982] Deveney, James K. "Ruled function fields." Proc. Amer. Math. Soc. 86 (1982), no. 2, 213–215. DOI: 10.2307/2043383. MR: 0667276.