  We establish a criterion for when an abelian extension of
infinite-dimensional Lie algebras integrates to a corresponding Lie group
extension $\hat{G}$ of $G$ by $A$, where $G$ is a connected, simply connected
Lie group and $A$ is a quotient of its Lie algebra by some discrete subgroup.
When $G$ is non-simply connected, the kernel $A$ is replaced by a central
extension $\hat{A}$ of $\pi_1(G)$ by $A$.
