We present near-infrared photometry (J band) and low-resolution optical spectroscopy (600-1000 nm) for one of the faintest substellar member candidates in the young σ Orionis cluster, S Ori 47 (I=20.53; Béjar, Zapatero Osorio, & Rebolo). Its very red (I-J) = 3.3+/-0.1 color and its optical spectrum allow us to classify S Ori 47 as an L1.5-type object that fits the low-luminosity end of the cluster photometric and spectroscopic sequences. It also displays atmospheric features indicative of low gravity, such as weak alkaline lines and hydride and oxide bands, consistent with the expectation for a very young object still undergoing gravitational collapse. Our data lead us to conclude that S Ori 47 is a true substellar member of the σ Orionis cluster. Additionally, we present the detection of Li I in its atmosphere, which provides an independent confirmation of youth and substellarity. Using current theoretical evolutionary tracks and adopting an age interval of 1-5 Myr for the σ Orionis cluster, we estimate the mass of S Ori 47 to be 0.015+/-0.005 M_solar, i.e., at the minimum mass for deuterium burning, which has been proposed as a definition for the boundary between brown dwarfs and giant planets. S Ori 47 could well be the result of a natural extension of the process of cloud fragmentation down to the deuterium-burning mass limit; a less likely alternative is that it has originated from a protoplanetary disk around a more massive cluster member and was later ejected from its orbit because of interacting effects within this rather sparse (~12 objects pc^-3) young cluster. The study of this object serves as a guide for future deep searches for free-floating objects with planetary masses.