The unusual helium-rich (type Ib) supernova SN 2005E is distinguished from all supernovae hitherto observed by its faint and rapidly fading light curve, prominent calcium lines in late-phase spectra and lack of any mark of recent star formation near the supernova location. These properties are claimed to be explained by a helium detonation in a thin surface layer of an accreting white dwarf. Here we report that the observed properties of SN 2005cz, which appeared in an elliptical galaxy, resemble those of SN 2005E. We argue that these properties are best explained by a core-collapse supernova at the low-mass end (8-12 solar masses) of the range of massive stars that explode. Such a low-mass progenitor lost its hydrogen-rich envelope through binary interaction, had very thin oxygen-rich and silicon-rich layers above the collapsing core, and accordingly ejected a very small amount of radioactive (56)Ni and oxygen. Although the host galaxy NGC 4589 is an elliptical, some studies have revealed evidence of recent star-formation activity, consistent with the core-collapse model.