This paper proves that the simultaneous lattice packing-covering constant of an octahedron is $7/6$. In other words, $7/6$ is the smallest positive number $r$ such that for every octahedron $O$ centered at the origin there is a lattice $Λ$ such that $O+Λ$ is a packing in $\mathbb{E}^3$ and $rO+Λ$ is a covering of $\mathbb{E}^3$.