An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting, i.e. Dixmier-Douady, is in $\operatorname {H}^2(X;\mathbb {Z} )\cup \operatorname {H}^1(X;\mathbb {Z} )\subset \operatorname {H}^3(X;\mathbb {Z} )$ and secondly that the 2- part is trivialized on the total space of the fibration. One of the features of this special case is that the corresponding Azumaya bundle can be refined to a bundle of smoothing operators. The topological and the analytic index of a projective family of elliptic operators associated with the smooth Azumaya bundle both take values in twisted $K$-theory of the parameterizing space and the main result is the equality of these two notions of index. The twisted Chern character of the index is then computed by a variant of Chern-Weil theory.