  The unusual helium-rich (type Ib) supernova SN 2005E is distinguished from
any supernova 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 (Perets et al. 2010). Here we report on observations of SN 2005cz
appeared in an elliptical galaxy, whose observed properties resemble those of
SN 2005E in that it is helium-rich and unusually faint, fades rapidly, shows
much weaker oxygen emission lines than those of calcium in the well-evolved
spectrum. We argue that these properties are best explained by a core-collapse
supernova at the low-mass end ($8-12 M_{\odot}$) of the range of massive stars
that explode (Smartt 2009). Such a low mass progenitor had lost its
hydrogen-rich envelope through binary interaction, having very thin oxygen-rich
and silicon-rich layers above the collapsing core, thus ejecting 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 (Zhang et al. 2008), consistent with the core-collapse scenario.
