Via optical and near-infrared imaging we have discovered an extremely red, low-luminosity population of free-floating objects in the young, nearby stellar cluster around the multiple, massive star σ Orionis. The proximity (352 pc), youth (1-5 Myr), and low internal extinction make this cluster an ideal site to explore the substellar domain from the hydrogen burning limit (~ 75 MJup) down to a few MJup. Optical and near-infrared low-resolution spectroscopy of three of our objects confirm the very cool spectral energy distribution (atmospheric effective temperatures of 2200-1700 K) expected for cluster members with masses in the range of 5-15 MJup. Like the planets of the Solar System, these objects are unable to sustain stable nuclear burning in their interiors, but in contrast they are not bound to stars. This new kind of isolated giant planet, which apparently forms in a timescale of less than a few million years, offers a challenge to our understanding of the formation processes of planetary mass objects.