  Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$c,d\in\dbN$. Let $b_{c,d}\ge 1$ be the smallest integer such that for any two
$p$-divisible groups $H$ and $H^\prime$ over $k$ of codimension $c$ and
dimension $d$ the following assertion holds: If $H[p^{b_{c,d}}]$ and
$H^\prime[p^{b_{c,d}}]$ are isomorphic, then $H$ and $H^\prime$ are isogenous.
We show that $b_{c,d}=\lceil{cd\over {c+d}}\rceil$. This proves Traverso's
isogeny conjecture for $p$-divisible groups over $k$.
