We report evidence for a well-defined quantum Hall effect in the field-induced spin-density-wave states of the organic superconductor (TMTSF)2ReO4. Hall resistance plateaus at h/2e2 and h/4e2 were clearly identified and correspond to the n=1 and n=2 states of the standard quantized nesting model. Magnetic breakdown is proposed as a tentative interpretation for the change in behavior of the Hall voltage occurring around 16 T and the existence of fast magnetoresistance oscillations. It is found that even in the semiconducting n=0 phase the absolute value of xy remains of the order of h/e2. A tentative explanation for this effect in terms of incipient field-induced localization is proposed.