A model for consistently describing the entire Ar exciton series is presented. Making use of a reliable approximation of the band structure in the whole first Brillouin zone, as provided by appropriately chosen Slater-Koster parameters, is an essential part of our approach. Its dominating effect on the final results is demonstrated by performing comparative studies based on several simpler band models. In fact, the energy defect of the n=1 exciton can to a large extent be understood as a consequence of the large-k variation of the bands. Our model, furthermore, makes allowance for the spin-orbit coupling of the valence bands, the exchange interaction and the dielectric function of the electron-hole potential. Without using any free parameter, the calculated Ar exciton series is in excellent agreement with the experimental binding energies.