  For an infinite dimensional Lie group $G$ modelled on a locally convex Lie
algebra $\mathfrak{g}$, we prove that every smooth projective unitary
representation of $G$ corresponds to a smooth linear unitary representation of
a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth
structure on $G^{\sharp}$.) For infinite dimensional Lie groups $G$ which are
1-connected, regular, and modelled on a barrelled Lie algebra $\mathfrak{g}$,
we characterize the unitary $\mathfrak{g}$-representations which integrate to
$G$. Combining these results, we give a precise formulation of the
correspondence between smooth projective unitary representations of $G$, smooth
linear unitary representations of $G^{\sharp}$, and the appropriate unitary
representations of its Lie algebra $\mathfrak{g}^{\sharp}$.
