Type Ia supernovae are thought to occur when a white dwarf made of carbon and oxygen accretes sufficient mass to trigger a thermonuclear explosion1. The accretion could be slow, from an unevolved (main-sequence) or evolved (subgiant or giant) star2,3 (the single-degenerate channel), or rapid, as the primary star breaks up a smaller orbiting white dwarf3,4 (the double-degenerate channel). A companion star will survive the explosion only in the single-degenerate channel5. Both channels might contribute to the production of type Ia supernovae6,7, but the relative proportions of their contributions remain a fundamental puzzle in astronomy. Previous searches for remnant companions have revealed one possible case for SN 1572 (refs 8, 9), although that has been questioned10. More recently, observations have restricted surviving companions to be small, main-sequence stars11,12,13, ruling out giant companions but still allowing the single-degenerate channel. Here we report the results of a search for surviving companions of the progenitor of SN 1006 (ref. 14). None of the stars within 4 arc minutes of the apparent site of the explosion is associated with the supernova remnant, and we can firmly exclude all giant and subgiant stars from being companions of the progenitor. In combination with previous results, our findings indicate that fewer than 20 per cent of type Ia supernovae occur through the single-degenerate channel.