This paper proposes a method to determine the virial coefficients of a substance from experimental density measurements. The method consists of a technique to establish the proper number of terms in the truncated series for a given set of data or to establish the proper data range for a given truncated series. The new methodology when applied to experimental (P-rho-T) data for argon provides results comparable to those reported previously in the literature using curve-fitting techniques or advanced graphical analyses. The proposed method uses additional information from independent measurements of molar mass to provide an objective criterion for selecting the number of terms in the virial expansion and/or the required number of experimental data to determine second and third virial coefficients accurately. The method uses mass densities (as produced by magnetic suspension densimeters, for example) and adjusts the fits of the data until the proper molecular mass results. The paper provides analyses of data from the literature and comparison to virial coefficients provided in the literature.