We present DES16C3cje, a low-luminosity, long-lived type II supernova (SN II) at redshift 0.0618, detected by the Dark Energy Survey (DES). DES16C3cje is a unique SN. The spectra are characterized by extremely narrow photospheric lines corresponding to very low expansion velocities of $lesssim1500$ km s$^{-1}$, and the light curve shows an initial peak that fades after 50 days before slowly rebrightening over a further 100 days to reach an absolute brightness of M$_rsim -15.5$ mag. The decline rate of the late-time light curve is then slower than that expected from the powering by radioactive decay of $^{56}$Co but is comparable to that expected from accretion power. Comparing the bolometric light curve with hydrodynamical models, we find that DES16C3cje can be explained by either i) a low explosion energy (0.11 foe) and relatively large $^{56}$Ni production of 0.075 M$_{odot}$ from a $sim15$ M$_{odot}$ red supergiant progenitor typical of other SNe II, or ii) a relatively compact $sim40$ M$_{odot}$ star, explosion energy of 1 foe, and 0.08 M$_{odot}$ of $^{56}$Ni. Both scenarios require additional energy input to explain the late-time light curve, which is consistent with fallback accretion at a rate of $sim0.5times{10^{-8}}$ M$_{odot}$ s$^{-1}$.
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