Over a hundred rocky planets orbiting Sun-like stars in very short orbital periods (<1 day) have been discovered by the Kepler mission. These planets, known as ultra-short-period (USP) planets, are unlikely to have attained their orbits in situ. Instead, they must have migrated in. Here we propose that these planets reach their current orbits by high-eccentricity migration. In a scaled-down version of the dynamics that may have been experienced by their high mass analog, the hot Jupiters, these planets reach high eccentricities via chaotic secular interactions with their companion planets and then undergo orbital circularization due to dissipation of tides raised on the planet. This proposal is motivated by the following observations: planetary systems observed by Kepler often contain several super-Earths with non-negligible eccentricities and inclinations, and possibly extending beyond AU distances; while only a small fraction of USP planets have known transiting companions, and none closely spaced, we argue that most of them should have companions at periods of ~10-50 days. The outer sibling planets, through secular chaos, can remove angular momentum from the inner most planet, originally at periods of ~5-10 days. When the latter reaches an eccentricity higher than 0.8, it is tidally captured by the central star and becomes an USP planet. This scenario naturally explains the observation that most USP planets have significantly more distant transiting companions compared to their counterparts at slightly longer periods (1-3 days), a feature un-accounted for in other proposed scenarios. Our model also predicts that USP planets should have: (i) spin-orbit angles, and inclinations relative to outer planets, in the range of ~10-50 degrees; (ii) several outer planetary companions extending to beyond AU distances, both of which may be tested by TESS and its follow-up observations.