Measurements of the electronic specific heat coefficient and of the limiting Debye temperature are reported for pure zinc and for two $\eta $phase alloys containing 2 at.% and 4 at.% silver in zinc, respectively. After a correction for electron-phonon enhancement the electronic specific heat coefficient for pure zinc differs by only a small percentage from the calculated value reported in the literature on the basis of a band calculation. The results for the alloys show a decreasing trend of the density of states at the Fermi level when silver is added to zinc. This is contrary to a prediction based on a rigid band approach. Hence, the results indicate a complete breakdown of the rigid band condition on alloying. The reasons for this are most likely associated with the influence of the $d$ band electrons or with charge distribution effects between solute and solvent atoms.