  We consider ensembles of real symmetric band matrices with entries drawn from
an infinite sequence of exchangeable random variables, as far as the symmetry
of the matrices permits. In general the entries of the upper triangular parts
of these matrices are correlated and no smallness or sparseness of these
correlations is assumed. It is shown that the eigenvalue distribution measures
still converge to a semicircle but with random scaling. We also investigate the
asymptotic behavior of the corresponding $\ell_2$-operator norms. The key to
our analysis is a generalisation of a classic result by de Finetti that allows
to represent the underlying probability spaces as averages of Wigner band
ensembles with entries that are not necessarily centred. Some of our results
appear to be new even for such Wigner band matrices.
