  The simplest type III seesaw model as originally proposed introduces one
lepton triplet. It thus contains four active neutrinos, two massive and two
massless at tree level. We determine the radiative masses that the latter
receive first at two loops. The masses are generally so tiny that they are
definitely excluded by the oscillation data, if the heavy leptons are not very
heavy, say, within the reach of LHC. To accommodate the data on masses, the
seesaw scale must be as large as the scale of grand unification. This indicates
that the most economical type III model would entail no new physics at low
energies beyond the tiny neutrino masses.
