The zodiacal dust bands were discovered by the Infrared Astronomical Satellite (IRAS). They form pairs of bright infrared (IR) emission strips that run along the ecliptic. Two of the three identified dust bands, known as beta and gamma, were previously linked to the recent asteroid breakups that produced groups of asteroid-sized fragments, so-called asteroid families, near orbits of (832) Karin and (490) Veritas. The origin of the third, near-ecliptic alpha dust band has been unknown until now. Here we report a discovery of a recent breakup of a >20-km-diameter asteroid near the alpha band's originally suspected source location in the Themis family. Numerical modeling and our new Spitzer observations of the alpha-band thermal emission allowed us to show that the discovered breakup is the source of alpha-band dust particles. This result has important implications for our understanding of the contribution of asteroid sources to the zodiacal cloud and the origin of circumstellar debris disk in general. It may also help to explain the observed cometary activity of asteroid (7968) Elst-Pizzaro because this object, one of the identified fragments of the recent breakup in the Themis family, is probably unusually young.