The classical third virial coefficient of argon has been computed for temperatures between 113.15 K and 1223.15 K using an ab initio three-body potential. The results agree with the virial coefficients extracted directly from experimental data to within about 5% for temperatures between 220 K and 420 K, theory almost touching the experimental error bars. For lower temperatures the discrepancies become larger, but results from different experiments are also significantly apart. Our calculated values agreed very well with those of Dymond and Alder [J. Chem. Phys. 54, 3472 (1971)] determined by fitting experimental data with theoretical constraints. The equation of state obtained using the computed second and third virial coefficients reproduced experimental PV data over a wide range of pressure.