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 of1500 km s-1, and the light curve shows an initial peak that fades after 50 d before slowly rebrightening over a further 100 d to reach an absolute brightness of Mr ∼-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 56Co, 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 56Ni production of 0.075 M· from an ∼15 M· red supergiant progenitor typical of other SNe II, or (ii) a relatively compact ∼40 M· star, explosion energy of 1 foe, and 0.08 M· of 56Ni. Both scenarios require additional energy input to explain the late-time light curve, which is consistent with fallback accretion at a rate of ∼0.5 × 10-8 M· s-1.