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 ensuremathlesssim1500 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 M_r ensuremathsim -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_ensuremathodot from an ensuremathsim15 M_ensuremathodot red supergiant progenitor typical of other SNe II, or (II) a relatively compact ensuremathsim40 M_ensuremathodot star, explosion energy of 1 foe, and 0.08 M_ensuremathodot 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 ensuremathsim0.5 texttimes 10^-8 M_ensuremathodot s^-1.