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Melting of the world's major ice sheets can affect human and environmental conditions by contributing to sea-level rise. In July 2012, an historically rare period of extended surface melting was observed across almost the entire Greenland ice sheet, raising questions about the frequency and spatial extent of such events. Here we show that low-level clouds consisting of liquid water droplets ('liquid clouds'), via their radiative effects, played a key part in this melt event by increasing near-surface temperatures. We used a suite of surface-based observations, remote sensing data, and a surface energy-balance model. At the critical surface melt time, the clouds were optically thick enough and low enough to enhance the downwelling infrared flux at the surface. At the same time they were optically thin enough to allow sufficient solar radiation to penetrate through them and raise surface temperatures above the melting point. Outside this narrow range in cloud optical thickness, the radiative contribution to the surface energy budget would have been diminished, and the spatial extent of this melting event would have been smaller. We further show that these thin, low-level liquid clouds occur frequently, both over Greenland and across the Arctic, being present around 30-50 per cent of the time. Our results may help to explain the difficulties that global climate models have in simulating the Arctic surface energy budget, particularly as models tend to under-predict the formation of optically thin liquid clouds at supercooled temperatures--a process potentially necessary to account fully for temperature feedbacks in a warming Arctic climate.