  Let $G$ be a compact connected Lie group and $k \in H^4(BG,\mathbb{Z})$ a
cohomology class. The String 2-group $G_k$ is the central extension of $G$ by
the 2-group $[\ast/U(1)]$ classified by $k$. It has a close relationship to the
level $k$ extension of the loop group $LG$. We compute the Drinfel'd centre of
$G_k$ as a smooth 2-group. When $G$ is semisimple, we prove that the Drinfel'd
centre is equal to the invertible part of the category of positive energy
representations of $LG$ at level $k$ (as long as we exclude factors of $E_8$ at
level 2).
