Bedrock river incision drives the development of much of Earth's surface topography(1-3), and thereby shapes the structure of mountain belts(4) and modulates Earth's habitability through its effects on soil erosion(5), nutrient fluxes(6) and global climate(7). Although it has long been expected that river incision rates should depend strongly on precipitation rates, quantifying the effects of precipitation rates on bedrock river incision rates has proved difficult, partly because river incision rates are difficult to measure and partly because non-climatic factors can obscure climatic effects at sites where river incision rates have been measured(8,9). Here we present measurements of river incision rates across one of Earth's steepest rainfall gradients, which show that precipitation rates do indeed influence long-term bedrock river incision rates. We apply a widely used empirical law for bedrock river incision(3,9-11) to a series of rivers on the Hawaiian island of Kaua'i, where mean annual precipitation ranges from 0.5 metres to 9.5 metres (ref. 12)-over 70 per cent of the global range(13)-and river incision rates averaged over millions of years can be inferred from the depth of river canyons and the age of the volcanic bedrock. Both a time-averaged analysis and numerical modelling of transient river incision reveal that the long-term efficiency of bedrock river incision across Kaua'i is positively correlated with upstream-averaged mean annual precipitation rates. We provide theoretical context for this result by demonstrating that our measurements are consistent with a linear dependence of river incision rates on stream power, the rate of energy expenditure by the flow on the riverbed. These observations provide rare empirical evidence for the long-proposed coupling between climate and river incision, suggesting that previously proposed feed-backs among topography, climate and tectonics may occur.