The near-Earth object (NEO) population, which mainly consists of fragments from collisions between asteroids in the main asteroid belt, is thought to include contributions from short-period comets as well. One of the most promising NEO candidates for a cometary origin is near-Earth asteroid (3552) Don Quixote, which has never been reported to show activity. Here we present the discovery of cometary activity in Don Quixote based on thermal-infrared observations made with the Spitzer Space Telescope in its 3.6 and 4.5 {μ}m bands. Our observations clearly show the presence of a coma and a tail in the 4.5 {μ}m but not in the 3.6 {μ}m band, which is consistent with molecular band emission from CO2. Thermal modeling of the combined photometric data on Don Quixote reveals a diameter of 18.4 (-0.4/+0.3) km and an albedo of 0.03 (-0.01/+0.02), which confirms Don Quixote to be the third-largest known NEO. We derive an upper limit on the dust production rate of 1.9 kg s^-1 and derive a CO2 gas production rate of (1.1+-0.1)10^26 molecules s^-1. Spitzer IRS spectroscopic observations indicate the presence of fine-grained silicates, perhaps pyroxene rich, on the surface of Don Quixote. Our discovery suggests that CO2 can be present in near-Earth space over a long time. The presence of CO2 might also explain that Don Quixote's cometary nature remained hidden for nearly three decades. ; Comment: 40 pages, 8 figures, accepted by ApJ