  Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of
$F=\mathrm{Frac} W(k)$ contained in some $F(\mu_{p^r})$. Using crystalline
Dieudonn\'e theory, we provide a classification of $p$-divisible groups over
$\mathscr{O}_K$ in terms of finite height $(\varphi,\Gamma)$-modules over
$\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of
(a special case of) the theory of Kisin--Ren, our construction gives an
independent proof and allows us to recover the Dieudonn\'e crystal of a
$p$-divisible group from the Wach module associated to its Tate module by
Berger--Breuil or by Kisin--Ren.
