Remark 2

$\mathrm{ht}(p) = 1$ and $\mathrm{ht}(q) \leqslant 1$. Then $A_p$ is a discrete valuation ring; let $\pi$ be a uniformizing element of $A_p$. Since $B_q$ is faithfully flat over $A_p$, the element $\pi 1_{B_q}$ of $B_q$ is not a zero divisor, so that the local ring $B_q/\pi B_q$ is of dimension 0 (VIII, § 3, No. 1, Cor. 2 of Prop. 1). It is reduced, since it is a ring of fractions of the reduced ring $\kappa(p) \otimes_A B$, and consequently it is a field. Therefore $B_q$ is a discrete valuation ring (VI, § 1, No. 4, Prop. 2), hence integrally closed.