  We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative
bundle gerbes associated to any principal $G$-bundle with connection and a
class in $H^4(BG, \ZZ)$ for a compact semi-simple Lie group $G$. The
Chern-Simons bundle 2-gerbe realises differential geometrically the
Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten
correspondence between three dimensional Chern-Simons functionals and
Wess-Zumino-Witten models associated to the group $G$. We do this by
introducing a lifting to the level of bundle gerbes of the natural map from
$H^4(BG, \ZZ)$ to $H^3(G, \ZZ)$. The notion of a multiplicative bundle gerbe
accounts geometrically for the subtleties in this correspondence for non-simply
connected Lie groups. The implications for Wess-Zumino-Witten models are also
discussed.
