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The melting of glaciers and ice caps accounts for about one-third of current sea-level rise 1-3 , exceeding the mass loss from the more voluminous Greenland or Antarctic Ice Sheets 3,4 . The Arctic archipelago of Svalbard, which hosts spatial climate gradients that are larger than the expected temporal climate shifts over the next century 5,6 , is a natural laboratory to constrain the climate sensitivity of glaciers and predict their response to future warming. Here we link historical and modern glacier observations to predict that twenty-first century glacier thinning rates will more than double those from 1936 to 2010. Making use of an archive of historical aerial imagery 7 from 1936 and 1938, we use structure-from-motion photogrammetry to reconstruct the three-dimensional geometry of 1,594 glaciers across Svalbard. We compare these reconstructions to modern ice elevation data to derive the spatial pattern of mass balance over a more than 70-year timespan, enabling us to see through the noise of annual and decadal variability to quantify how variables such as temperature and precipitation control ice loss. We find a robust temperature dependence of melt rates, whereby a 1 °C rise in mean summer temperature corresponds to a decrease in area-normalized mass balance of -0.28 m yr -1 of water equivalent. Finally, we design a space-for-time substitution 8 to combine our historical glacier observations with climate projections and make first-order predictions of twenty-first century glacier change across Svalbard., (© 2022. The Author(s), under exclusive licence to Springer Nature Limited.)