In this second paper on the entire virial region of the relaxed fossil cluster RXJ1159+5531, we present a hydrostatic analysis of the hot intracluster medium (ICM). For a model consisting of ICM, stellar mass from the central galaxy (BCG), and an NFW dark matter (DM) halo, we obtain good descriptions of the projected radial profiles of ICM emissivity and temperature. The BCG stellar mass is clearly detected with M_star/L_K = 0.61 +/- 0.11 solar, consistent with stellar population synthesis models for a Milky-Way IMF. We obtain a halo concentration, c_200 =8.4 +/- 1.0, and virial mass, M_200 = 7.9 +/- 0.6 x 10^13 M_sun. For its mass, the inferred concentration is larger than most relaxed halos produced in cosmological simulations with Planck parameters, consistent with RXJ1159+5531 forming earlier than the general halo population. The baryon fraction at r_200, f_b,200 = 0.134 +/- 0.007, is slightly below the Planck value (0.155) for the universe. When we account for the stellar baryons associated with non-central galaxies and the uncertain intracluster light, f_b,200 increases by ~0.015, consistent with the cosmic value. Performing our analysis in the context of MOND still requires a large DM fraction (85.0% +/- 2.5% at r=100 kpc) similar to that obtained using the standard Newtonian approach. The detection of a plausible stellar BCG mass component distinct from the NFW DM halo in the total gravitational potential suggests that ~10^14 M_sun represents the mass scale above which dissipation is unimportant in the formation of the central regions of galaxy clusters. (Abridged)
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