The superfluidity of He-4 films adsorbed on the atomically flat surfac
e of graphite, preplated with KD to tune the surface binding potential
, has been studied using a torsional oscillator. The superfluidity of
a single uniform fluid layer of He-4 shows an intrinsic coverage depen
dent suppression, while the fluid bilayer is fully superfluid at T = 0
. The contribution of nonvortex excitations in the film to the normal
density shows a strong dependence on coverage, arising from the atomic
layering of the film.