The superfluidity of 4 He films adsorbed on the atomically flat surface of graphite, preplated with HD to tune the surface binding potential, has been studied using a torsional oscillator. The superfluidity of a single uniform fluid layer of 4 He shows an intrinsic coverage dependent 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.[S0031-9007 (98)06517-X] PACS numbers: 67.70. + n, 67.40.Db, 67.40.Kh A thin 4 He film on a planar substrate is a paradigm two dimensional Bose system. It is predicted to undergo a superfluid phase transition due to the unbinding of vortex-antivortex pairs [1], with a discontinuous jump in the superfluid density obeying a universal scaling relation with the transition temperature [2]; r s ͑T c ͒͞T c 2m 2 k B ͞ph 2 . This was first verified by torsional oscillator studies of films adsorbed on a Mylar sheet [3].Since such an atomically rough substrate provides a heterogeneous binding potential for the adsorbed 4 He atoms, there is a threshold coverage, referred to as the "inert" layer, before superfluidity is observed. The simplest picture is that an amorphous solid 4 He coating of the surface is required in order to screen the disordered substrate potential, before subsequent 4 He atoms are delocalized and can undergo a superfluid transition [4].More recently there has been renewed interest in films adsorbed on the atomically flat surface of graphite, which provides an essentially uniform binding potential, resulting in a 4 He film that, by contrast, displays distinct atomic layering. Evidence for such layering comes from vapor pressure isotherms [5], heat capacity [6], and third sound measurements [5], as well as first principles calculations of the film structure [7]. This layered structure influences the development of superfluidity in the film, as first shown by Crowell and Reppy [8].This Letter discusses (i) the superfluidity of a fluid monolayer, (ii) the properties of a superfluid bilayer, and (iii) the dependence of the nonvortex excitations in the film on its structure. We have made a systematic investigation of the effect of tuning the substrate potential by preplating the graphite surface with hydrogen deuteride (HD) on the superfluid response. The number of solid 4 He layers that can form is reduced to one for a bilayer [5] or trilayer preplating, and zero for the thick preplating film we have investigated. By contrast, two 4 He layers will solidify on bare graphite. In these systems we are able to study the superfluid transition for a single fluid layer, which for the thick preplating corresponds to "submonolayer superfluidity." For all three preplatings we find 2D condensation for fluid coverages ,3.5 nm 22 and clear evidence at higher densities for a coverage dependent suppression of superfluidity in the uniform fluid layer. This latter effect is quite distinct from the "inert layer" found on hetero...