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 ~27Mpc. It reached a peak absolute magnitude of M_r_=-12.66+/-0.20mag, 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 0.5mag, the peak absolute magnitude would be M_r_=-13.78+/-0.20mag-still consistent with being the lowest-luminosity SN. Optical spectra of SN 2021fcg taken at 37 and 65 days post-maximum show strong [CaII], CaII, and NaID emission and several weak [FeII] emission lines. The [CaII] emission in the two spectra has extremely low velocities of ~1300 and 1000km/s, respectively. The spectra very closely resemble those of the very low luminosity Type Iax supernovae SN2008ha, SN2010ae, and SN2019gsc taken at similar phases. The peak bolometric luminosity of SN2021fcg is ~2.5_-0.3_^+1.5^x10^40^erg/s, which is a factor of 3 lower than that for SN2008ha. The bolometric lightcurve of SN2021fcg is consistent with a very low ejected nickel mass (M_Ni_~0.8_-0.5_^+0.4^x10^-3^M_{sun}_). The low luminosity and nickel mass of SN2021fcg pose a challenge to the picture that low-luminosity SNe Iax originate from deflagrations of near-Mch hybrid carbon-oxygen-neon white dwarfs. Instead, the merger of a carbon-oxygen and oxygen-neon white dwarf is a promising model to explain SN2021fcg.