Supergiant stars such as Betelgeuse have very extended atmospheres, the properties of which are poorly understood.Alfven waves [1-4], acoustic waves [1,2,5-7] and radial pulsations  have all been suggested as likely mechanisms for elevating these atmospheres and driving the massive outflows of gas seen in these stars: such mechanisms would heat the atmosphere from below, and there are indeed observations showing that Betelgeuse's extended atmosphere is hotter than the underlying photosphere [9,10]. Here we report radio observations of Betelgeuse that reveal the temperature structure of the extended atmosphere from two to seven times the photospheric radius. Close to the star, we find that the atmosphere has an irregular structure, and a temperature (3,450 +/- 850 K) consistent with the photospheric temperature but much lower than that of gas in the same region probed by optical and ultraviolet observations . This cooler gas decreases steadily in temperature with radius, reaching 1,370 +/- 330 K by seven stellar radii. The cool gas coexists with the hot chromospheric gas, but must be much more abundant as it dominates the radio emission. Our results suggest that a few inhomogeneously distributed large convective cells (which are widely believed [11-16] to be present in such stars) are responsible for lifting the cooler photospheric gas into the atmosphere; radiation pressure on dust grains that condense from this gas may then drive Betelgeuse's outflow.