Sympathetic cooling with ultracold atoms(1) and atomic ions(2) 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(3). Here, we use ultracold atoms to sympathetically cool the vibrations of a Si3N4 nanomembrane(4,5), the mass of which 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(6,7) and is enhanced by placing the membrane in an optical cavity(8). We observe cooling of the membrane vibrations from room temperature to 650 +/- 230 mK, exploiting the large atom-membrane cooperativity(9) of our hybrid optomechanical system(10,11). With technical improvements, our scheme could provide ground-state cooling and quantum control of low-frequency oscillators such as nanomembranes or levitated nanoparticles(12,13), in a regime where purely optomechanical techniques cannot reach the ground state(8,9).