We present the discovery by optical and near-infrared imaging of an extremely red, low-luminosity population of isolated objects in the young, nearby stellar cluster around the multiple, massive star σ Orionis. The proximity (352 parsecs), youth (1 million to 5 million years), and low internal extinction make this cluster an ideal site to explore the substellar domain from the hydrogen mass limit down to a few Jupiter masses. Optical and near-infrared low-resolution spectroscopy of three of these objects confirms the very cool spectral energy distribution (atmospheric effective temperatures of 1700 to 2200 kelvin) expected for cluster members with masses in the range 5 to 15 times that of Jupiter. 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 on time scales of less than a few million years, offers a challenge to our understanding of the formation processes of planetary mass objects.