  New unique continuation principles or uncertainty relations for linear
combinations of eigenfunctions of the Laplacian on cubes from \cite{EgidiV-20}
are used to prove a Wegner estimate for alloy type models along the lines of
previously known proofs, e.~g.~\cite{CombesHK-07}. Due to the strong input we
merely have to assume that the single site potential is lower bounded by a
characteristic function of a set of positive measure. We obtain a Wegner
estimate with optimal volume dependence at all energies, and localisation near
the minimum of the spectrum, even for some non-stationary random potentials.
