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 $650pm 230$ 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.
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