A new method is introduced for making X-ray mass determinations of spherical clusters of galaxies. Treating the distribution of gravitating matter as piecewise constant and the cluster atmosphere as piecewise isothermal, X-ray spectra of a hydrostatic atmosphere are determined up to a single overall normalizing factor. In contrast to more conventional approaches, this method relies on the minimum of assumptions, apart from the conditions of hydrostatic equilibrium and spherical symmetry. The method has been implemented as an XSPEC mixing model called CLMASS, which was used to determine masses for a sample of nine relaxed X-ray clusters. Compared to conventional mass determinations, CLMASS provides weak constraints on values of M_500, reflecting the quality of current X-ray data for cluster regions beyond r_500. At smaller radii, where there are high quality X-ray spectra inside and outside the radius of interest to constrain the mass, CLMASS gives confidence ranges for M_2500 that are only moderately less restrictive than those from more familiar mass determination methods. The CLMASS model provides some advantages over other methods and should prove useful for mass determinations in regions where there are high quality X-ray data.
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