Most near-Earth objects came from the asteroid belt and drifted via non-gravitational thermal forces into resonant escape routes that, in turn, pushed them onto planet-crossing orbits. Models predict that numerous asteroids should be found on orbits that closely approach the Sun, but few have been seen. In addition, even though the near-Earth-object population in general is an even mix of low-albedo (less than ten per cent of incident radiation is reflected) and high-albedo (more than ten per cent of incident radiation is reflected) asteroids, the characterized asteroids near the Sun typically have high albedos. Here we report a quantitative comparison of actual asteroid detections and a near-Earth-object model (which accounts for observational selection effects). We conclude that the deficit of low-albedo objects near the Sun arises from the super-catastrophic breakup (that is, almost complete disintegration) of a substantial fraction of asteroids when they achieve perihelion distances of a few tens of solar radii. The distance at which destruction occurs is greater for smaller asteroids, and their temperatures during perihelion passages are too low for evaporation to explain their disappearance. Although both bright and dark (high- and low-albedo) asteroids eventually break up, we find that low-albedo asteroids are more likely to be destroyed farther from the Sun, which explains the apparent excess of high-albedo near-Earth objects and suggests that low-albedo asteroids break up more easily as a result of thermal effects.