  Sympathetic cooling with ultracold atoms and atomic ions enables ultralow
temperatures in systems where direct laser or evaporative cooling is not
possible. It has so far been limited to the cooling of other microscopic
particles, with masses up to $90$ times larger than that of the coolant atom.
Here we use ultracold atoms to sympathetically cool the vibrations of a
Si$_3$N$_4$ nanomembrane, whose mass exceeds that of the atomic ensemble by a
factor of $10^{10}$. The coupling of atomic and membrane vibrations is mediated
by laser light over a macroscopic distance and enhanced by placing the membrane
in an optical cavity. We observe cooling of the membrane vibrations from room
temperature to $650\pm 230$ mK, exploiting the large atom-membrane
cooperativity of our hybrid optomechanical system. Our scheme enables
ground-state cooling and quantum control of low-frequency oscillators such as
nanomembranes or levitated nanoparticles, in a regime where purely
optomechanical techniques cannot reach the ground state.
