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The temperature of the Sun's outer atmosphere (the corona) exceeds that of the solar surface by about two orders of magnitude, but the nature of the coronal heating mechanisms has long been a mystery. The corona is a magnetically dominated environment, consisting of a variety of plasma structures including X-ray bright points, coronal holes and coronal loops. The latter are closed magnetic structures that occur over a range of scales and are anchored at each end in the solar surface. Large-scale regions of diffuse emission are made up of many long coronal loops. Here we present X-ray observations of the diffuse corona from which we deduce its likely heating mechanism. We find that the observed variation in temperature along a loop is highly sensitive to the spatial distribution of the heating. From a comparison of the observations and models we conclude that uniform heating gives the best fit to the loop temperature distribution, enabling us to eliminate previously suggested mechanisms of low-lying heating near the footpoints of a loop. Our findings favour turbulent breaking and reconnection of magnetic field lines as the heating mechanism of the diffuse solar corona.