  We show that there is a $\beta$-model of second-order arithmetic in which the
choice scheme holds, but the dependent choice scheme fails for a
$\Pi^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a
corollary that the Reflection Principle, stating that every formula reflects to
a transitive set, can fail in models of ${\rm ZFC}^-$. This work is a
rediscovery by the first two authors of a result obtained by the third author.
