Supernovae are thought to arise from two different physical processes. The cores of massive, short-lived stars undergo gravitational core collapse and typically eject a few solar masses during their explosion. These are thought to appear as type Ib/c and type II supernovae, and are associated with young stellar populations. In contrast, the thermonuclear detonation of a carbon-oxygen white dwarf, whose mass approaches the Chandrasekhar limit, is thought to produce type Ia supernovae. Such supernovae are observed in both young and old stellar environments. Here we report a faint type Ib supernova, SN 2005E, in the halo of the nearby isolated galaxy, NGC 1032. The 'old' environment near the supernova location, and the very low derived ejected mass ( approximately 0.3 solar masses), argue strongly against a core-collapse origin. Spectroscopic observations and analysis reveal high ejecta velocities, dominated by helium-burning products, probably excluding this as a subluminous or a regular type Ia supernova. We conclude that it arises from a low-mass, old progenitor, likely to have been a helium-accreting white dwarf in a binary. The ejecta contain more calcium than observed in other types of supernovae and probably large amounts of radioactive (44)Ti.