The stellar initial mass function (IMF) describes the distribution in stellar masses produced from a burst of star formation. For more than 50 yr, the implicit assumption underpinning most areas of research involving the IMF has been that it is universal, regardless of time and environment. We measure the high-mass IMF slope for a sample of low-to-moderate redshift galaxies from the Galaxy and Mass Assembly survey. The large range in luminosities and galaxy masses of the sample permits the exploration of underlying IMF dependencies. A strong IMF-star formation rate dependency is discovered, which shows that highly star-forming galaxies form proportionally more massive stars (they have IMFs with flatter power-law slopes) than galaxies with low star formation rates. This has a significant impact on a wide variety of galaxy evolution studies, all of which rely on assumptions about the slope of the IMF. Our result is supported by, and provides an explanation for, the results of numerous recent explorations suggesting a variation of or evolution in the IMF.