  Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $\zeta_3$ a
primitive third root of unity. Recently, Lemmermeyer presented a conjecture
about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of
their normal closures $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$. The purpose
of this paper is to prove Lemmermeyer's conjecture.
