  We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and
$K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and
$3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn,
and proves the last remaining case of a conjecture of Lemmermeyer on the
$3$-class group of $K$.
