We combine results from imaging searches for substellar objects in the σ Orionis cluster and follow-up photometric and spectroscopic observations to derive a census of the brown dwarf population in a region of 847 arcmin2. We identify 64 very low mass cluster member candidates in this region. We have available three-color (I, Z, and J) photometry for all of them, spectra for 24 objects, and K photometry for 27% of our sample. These data provide a well-defined sequence in the I versus I-J and I versus I-K color-magnitude diagrams and indicate that the cluster exhibits little reddening despite its young age (~5 Myr). Using state-of-the-art evolutionary models, we derive a mass function from the low-mass stars (0.2 Msolar) across the complete brown dwarf domain (0.075 to 0.013 Msolar) and into the realm of free-floating planetary-mass objects (<=0.013 Msolar). We find that the mass spectrum (dN/dm)~m-α increases toward lower masses, with an exponent α=0.8+/-0.4. Our results suggest that planetary-mass isolated objects could be as common as brown dwarfs; both kinds of objects together would be as numerous as stars in the cluster. If the distribution of stellar and substellar masses in σ Orionis is representative of the Galactic disk, older and much lower luminosity free-floating planetary-mass objects with masses down to about 0.005 Msolar should be abundant in the solar vicinity, with a density similar to M-type stars.