  We prove that the local eigenvalue statistics for $d=1$ random band matrices
with fixed bandwidth and, for example, Gaussian entries, is given by a Poisson
point process and we identify the intensity of the process. The proof relies on
an extension of the localization bounds of Schenker \cite{schenker} and the
Wegner and Minami estimates. These two estimates are proved using averaging
over the diagonal disorder. The new component is a proof of the uniform
convergence and the smoothness of the density of states function. The limit
function, known to be the semicircle law with a band-width dependent error
\cite{bmp,dps,dl,mpk}, is identified as the intensity of the limiting Poisson
point process. The proof of these results for the density of states relies on a
new result that simplifies and extends some of the ideas used by Dolai,
Krishna, and Mallick \cite{dkm}. These authors proved regularity properties of
the density of states for random Schr\"odinger operators (lattice and
continuum) in the localization regime. The proof presented here applies to the
random Schr\"odinger operators on a class of infinite graphs treated by in
\cite{dkm} and extends the results of \cite{dkm} to probability measures with
unbounded support. The method also applies to fixed bandwidth RBM for $d=2,3$
provided certain localization bounds are known.
