= Solution
The <Going-up theorem> states: if $B$ is integral over $A$, $\mathfrak p_1\subseteq\mathfrak p_2$ are primes of $A$, and $\mathfrak q_1$ lies over $\mathfrak p_1$, then some prime $\mathfrak q_2\supseteq\mathfrak q_1$ lies over $\mathfrak p_2$.
Pass to $A/\mathfrak p_1\subseteq B/\mathfrak q_1$, which remains integral, and localize at the complement of $\mathfrak p_2/\mathfrak p_1$. The lying-over theorem supplies a prime of the localized upper ring over the maximal ideal of the localized lower ring. Contracting it to $B/\mathfrak q_1$, and then pulling it back to $B$, gives the required $\mathfrak q_2$.
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