SUPERNOVA 1993J in the spiral galaxy M81 is the brightest supernova since SN1987A and, like the latter, appears to be another 'peculiar' type II supernova. The available photometry1,2 of the supernova region before the explosion requires the presence of at least two supergiants (one of early spectral type and the other of late type), but the actual progenitor has yet to be identified. Here we show that the explosion of a late-type supergiant can explain the initial sharp peak in the supernova light curve, provided that the star had lost almost all of its hydrogen-rich envelope before the explosion. In our model, the secondary brightening of the supernova, approximately 10 days later, is then a consequence of the radioactive decay of Ni-56 and subsequently Co-56) produced in the explosion. The progenitor could have lost its hydrogen-rich envelope either in a strong stellar wind or, as seems more likely, through mass transfer to a companion star. In the latter case, the companion should reappear after the supernova photosphere has receded, the system having become a binary composed of a neutron star with a massive stellar companion.