Massive stars are very rare, but their extreme luminosities make them both the only type of young star we can observe in distant galaxies and the dominant energy sources in the Universe today. They form rarely because efficient radiative cooling keeps most star-forming gas clouds close to isothermal as they collapse, and this favours fragmentation into stars of one solar mass or lower. Heating of a cloud by accreting low-mass stars within it can prevent fragmentation and allow formation of massive stars, but the necessary properties for a cloud to form massive stars-and therefore where massive stars form in a galaxy-have not yet been determined. Here we show that only clouds with column densities of at least 1 gcm~(-2) can avoid fragmentation and form massive stars. This threshold, and the environmental variation of the stellar initial mass function that it implies, naturally explain the characteristic column densities associated with massive star dusters and the difference between the radial profiles of Ha and ultraviolet emission in galactic disks. The existence of a threshold also implies that the initial mass function should show detectable variation with environment within the Galaxy, that the characteristic column densities of clusters containing massive stars should vary between galaxies, and that star formation rates in some galactic environments may have been systematically underestimated.
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