We report intermediate-resolution spectroscopy in the range 625-775 nm of a faint (R approximately 19) very late type object, discovered in two CCD images taken near the center of the alpha-Per cluster. The spectrum shows strong absorption molecular bands, H-alpha emission, and KI absorption typical of late-M dwarfs; H-alpha varies conspicuously (W(lambda) approximately 2.4-0.8 nm) is a short time scale. The effective temperature and luminosity derived from the spectrum and R,I band photometry are consistent with those expected for an alpha-Per (age approximately 50 Myr) 0.07 M. substellar object. We argue that in our object the lithium resonance line at 670.8 nm should be detectable with higher resolution, higher signal-to-noise ratio data, and we propose that this line is a potentially powerful spectroscopic discriminant between very low mass stars and brown dwarfs close to the substellar mass limit.