  We describe an interesting relation between Lie 2-algebras, the Kac-Moody
central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie
2-algebra is a categorified version of a Lie algebra where the Jacobi identity
holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie
2-group is a categorified version of a Lie group. If $G$ is a simply-connected
compact simple Lie group, there is a 1-parameter family of Lie 2-algebras
$\mathfrak{g}_k$ each having $\mathrm{Lie}(G)$ as its Lie algebra of objects,
but with a Jacobiator built from the canonical 3-form on $G$. There appears to
be no Lie 2-group having $\mathfrak{g}_k$ as its Lie 2-algebra, except when $k
= 0$. Here, however, we construct for integral k an infinite-dimensional Lie
2-group whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. The objects of
this 2-group are based paths in $G$, while the automorphisms of any object form
the level-$k$ Kac-Moody central extension of the loop group of $G$. This
2-group is closely related to the $k$th power of the canonical gerbe over $G$.
Its nerve gives a topological group that is an extension of $G$ by
$K(\mathbb{Z},2)$. When $k = \pm 1$, this topological group can also be
obtained by killing the third homotopy group of $G$. Thus, when $G =
\mathrm{Spin}(n)$, it is none other than $\mathrm{String}(n)$.
