Abstract Primordial black holes may encode the conditions of the early universe, and may even constitute a significant fraction of cosmological dark matter. Their existence has yet to be established. However, black holes with masses below ∼1M⊙ cannot form as an endpoint of stellar evolution, so the detection of even one such object would be a smoking gun for new physics, and would constitute evidence that at least a fraction of the dark matter consists of primordial black holes. Gravitational wave detectors are capable of making a definitive discovery of this kind by detecting mergers of light black holes. But since the merger rate depends strongly on the shape of the black hole mass function, it is difficult to determine the potential for discovery or constraint as a function of the overall abundance of black holes. Here, we directly maximize and minimize the merger rate to connect observational results to the actual abundance of observable objects. We show that LIGO can discover mergers of light primordial black holes within the next decade even if such black holes constitute only a very small fraction of dark matter. A single merger event involving such an object would (i) provide conclusive evidence of new physics, (ii) establish the nature of some fraction of dark matter, and (iii) probe cosmological history at scales far beyond those observable today.