Trapped quantum gases can be cooled to impressively low temperatures, but it is unclear whether their entropy is low enough to realize phenomena such as d-wave superconductivity and magnetic ordering. Estimated critical entropies per particle for quantum magnetic ordering are \( \sim 0.3k_B\) and \( \sim 0.03k_B\) for bosons in three- and two-dimensional lattices, respectively, with similar values for Néel ordering of lattice-trapped Fermi gases. Here we report reliable single-shot temperature measurements of a degenerate Rb gas by imaging the momentum distribution of thermalized magnons, which are spin excitations of the atomic gas. We record average temperatures fifty times lower than the Bose–Einstein condensation temperature, indicating an entropy per particle of \( \sim 0.001 k_B\) at equilibrium, nearly two orders of magnitude lower than the previous best in a dilute atomic gas, and well below the critical entropy for antiferromagnetic ordering of a Bose–Hubbard system. The magnons can reduce the temperature of the system by absorbing energy during thermalization and by enhancing evaporative cooling, allowing the production of low-entropy gases in deep traps.