  We present the discovery of ZTF 21aaoryiz/SN 2021fcg -- an extremely
low-luminosity Type Iax supernova. SN 2021fcg was discovered by the Zwicky
Transient Facility in the star-forming galaxy IC0512 at a distance of $\approx$
27 Mpc. It reached a peak absolute magnitude of $M_{r} =$ $-12.66\pm0.20$ mag,
making it the least luminous thermonuclear supernova discovered to date. The
E(B-V) contribution from the underlying host galaxy is unconstrained. However,
even if it were as large as 0.5 mag, the peak absolute magnitude would be
$M_{r} = -13.78\pm0.20$ mag -- still consistent with being the lowest
luminosity SN. Optical spectra of SN 2021fcg taken at 37 and 65 days post
maximum show strong [Ca II], Ca II and Na I D emission and several weak [Fe II]
emission lines. The [Ca II] emission in the two spectra has extremely low
velocities of $\approx 1300$ and $1000$ km s$^{-1}$ respectively. The spectra
very closely resemble those of the very low luminosity Type Iax supernovae SN
2008ha, SN 2010ae and SN 2019gsc taken at similar phases. The peak bolometric
luminosity of SN 2021fcg is $\approx$ $2.5^{+1.5}_{-0.3}\times10^{40}$ erg
s$^{-1}$ which is a factor of three lower than that for SN 2008ha. The
bolometric lightcurve of SN 2021fcg is consistent with a very low ejected
nickel mass (M$_{\rm{Ni}} \approx 0.8^{+0.4}_{-0.5}\times10^{-3}$ M$_{\odot}$).
The low luminosity and nickel mass of SN 2021fcg pose a challenge to the
picture that low luminosity SNe Iax originate from deflagrations of near
M$_{\rm{ch}}$ hybrid carbon-oxygen-neon white dwarfs. Instead, the merger of a
carbon-oxygen and oxygen-neon white dwarf is a promising model to explain SN
2021fcg.
