We have discovered a relatively large number of brown dwarf candidates in the Pleiades young open cluster as a result of a deep CCD IZ survey covering 1 deg^2 down to limiting magnitudes I ~22 and Z ~21. Follow-up infrared photometry and low/intermediate-resolution spectroscopy of seven of our candidates in the interval 20 > I > 17.8 allow us to investigate their membership in the Pleiades. The observed spectra show that these objects are very cool dwarfs (M6-M9). Five of them can be considered as cluster members on the basis of their spectroscopic and photometric properties. Given their low luminosity and according to current evolutionary models, we estimate their masses to range from roughly 80 M_Jup for the hottest object, down to 45 M_Jup for Roque 4, the coolest and faintest confirmed member. We observe some differences in the spectra of our least massive Pleiades brown dwarfs in comparison to field stars of very late spectral types and similar effective temperatures. They may be related to gravity effects and the presence of dust in the atmospheres. The brown dwarfs discovered in the Pleiades prove that the process of fragmentation of clouds extend well into the substellar realm, suggesting a rise in the inital mass function below the star-BD boundary.