A full-electron coupled-state treatment of positronium (Ps)-inert gas scattering is developed within the context of the frozen target approximation. Calculations are performed for Ps(1s) scattering by Ne and Ar in the impact energy range 0-40 eV using coupled pseudostate expansions consisting of nine and 22 Ps states. The purpose of the pseudostates is primarily to represent ionization of the Ps which is found to be a major process at the higher energies. First Born estimates of target excitation are used to complement the frozen target results. The available experimental data are discussed in detail. It is pointed out that the very low energy measurements (≤2 eV) correspond to the momentum transfer cross section σ_mom and not to the elastic cross section σ_el. Calculation shows that σ_mom and σ_el diverge very rapidly with increasing energy and consequently comparisons of the low-energy data with σ_el can be very misleading. Agreement between the calculations and the low-energy measurements of σ_mom, as well as higher energy (≥15 eV) beam measurements of the total cross section, is less than satisfactory. Results for Ps(1s) scattering by Kr and Xe in the static-exchange approximation are also presented.