The sensitivity of river discharge to climate-system warming is highly uncertain, and the processes that govern river discharge are poorly understood, which impedes climate-change adaptation. A prominent exemplar is the Colorado River, where meteorological drought and warming are shrinking a water resource that supports more than 1 trillion dollars of economic activity per year. A Monte Carlo simulation with a radiation-aware hydrologic model resolves the longstanding, wide disparity in sensitivity estimates and reveals the controlling physical processes. We estimate that annual mean discharge has been decreasing by 9.3% per degree Celsius of warming because of increased evapotranspiration, mainly driven by snow loss and a consequent decrease in reflection of solar radiation. Projected precipitation increases likely will not suffice to fully counter the robust, thermodynamically induced drying. Thus, an increasing risk of severe water shortages is expected.