The Alpha Magnetic Spectrometer experiment onboard the International Space Station has recently provided cosmic ray electron and positron data with unprecedented precision in the range from 0.5 to 350 GeV. The observed rise in the positron fraction at energies above 10 GeV remains unexplained, with proposed solutions ranging from local pulsars to TeV-scale dark matter. Here, we make use of this high quality data to place stringent limits on dark matter with masses below ~300 GeV, annihilating or decaying to leptonic final states, essentially independent of the origin of this rise. We significantly improve on existing constraints, in some cases by up to 2 orders of magnitude.