The layer-by-layer growth of solid argon films on graphite at T=10 K is studied using elastic neutron diffraction. The growth is characterized by individual layers with commensurate in-plane lattice constants. As the coverage is increased beyond two layers, evidence of the coexistence of ABC and ABA stacking is apparent, with the ABC sequence dominating as the film thickens. A continuous decrease in the Debye-Waller factor also occurs as the film thickness grows, indicating a crossover from two-dimensional to three-dimensional behavior. As the coverage is increased beyond about four nominal layers, there is evidence of bulk crystallite formation. The diffraction results are compared with equivalent measurements for methane films and with the recent computer simulations of Hruska and Phillips.