SONYC -Substellar Objects in Nearby Young Clusters- is a survey program to investigate the frequency and properties of substellar objects in nearby star-forming regions. We present a new imaging and spectroscopic survey conducted in the young (~1 Myr), nearby (~200 pc) star-forming region Lupus 3. Deep optical and near-infrared images were obtained with MOSAIC-II and NEWFIRM at the CTIO 4 m telescope, covering ~1.4 deg^2^ on the sky. The i-band completeness limit of 20.3 mag is equivalent to 0.009-0.02 M_{sun}_, for A_V_<=5. Photometry and 11-12 yr baseline proper motions were used to select candidate low-mass members of Lupus 3. We performed a spectroscopic follow-up of 123 candidates, using VIMOS at the Very Large Telescope, and we identify 7 probable members, among which 4 have spectral type later than M6.0 and T_eff_<=3000 K, i.e., are probably substellar in nature. Two of the new probable members of Lupus 3 appear underluminous for their spectral class and exhibit emission line spectrum with strong H_{alpha}_ or forbidden lines associated with active accretion. We derive a relation between the spectral type and effective temperature: T_eff_=(4120+/-175)-(172+/-26)xSpT, where SpT refers to the M spectral subtype between 1 and 9. Combining our results with the previous works on Lupus 3, we show that the spectral type distribution is consistent with that in other star-forming regions, as well as the derived star-to-brown dwarf ratio of 2.0-3.3. We compile a census of all spectroscopically confirmed low-mass members with spectral type M0 or later.