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 less than or similar to 1500 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 similar to 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 Co-56, 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 Ni-56 production of 0.075 M-circle dot from an similar to 15 M-circle dot red supergiant progenitor typical of other SNe II, or (ii) a relatively compact similar to 40 M-circle dot star, explosion energy of 1 foe, and 0.08 M-circle dot of Ni-56. Both scenarios require additional energy input to explain the late-time light curve, which is consistent with fallback accretion at a rate of similar to 0.5 x 10(-)(8) M-circle dot s(-1).