  Ultracold gases promise access to many-body quantum phenomena at convenient
length and time scales. However, it is unclear whether the entropy of these
gases is low enough to realize many phenomena relevant to condensed matter
physics, such as quantum magnetism. Here we report reliable single-shot
temperature measurements of a degenerate $^{87}$Rb gas by imaging the momentum
distribution of thermalized magnons, which are spin excitations of the atomic
gas. We record average temperatures as low as
$0.022(1)_\text{stat}(2)_\text{sys}$ times the Bose-Einstein condensation
temperature, indicating an entropy per particle, $S/N\approx0.001\, k_B$ at
equilibrium, that is well below the critical entropy for antiferromagnetic
ordering of a Bose-Hubbard system. The magnons themselves can reduce the
temperature of the system by absorbing energy during thermalization and by
enhancing evaporative cooling, allowing low-entropy gases to be produced within
deep traps.
