  If large extra dimensions exist in nature, supernova (SN) cores will emit
large fluxes of Kaluza-Klein gravitons, producing a cosmic background of these
particles with energies and masses up to about 100 MeV. Radiative decays then
give rise to a diffuse cosmic gamma-ray background with E_gamma < 100 MeV which
is well in excess of the observations if more than 0.5-1% of the SN energy is
emitted into the new channel. This argument complements and tightens the
well-known cooling limit from the observed duration of the SN1987A neutrino
burst. For two extra dimensions we derive a conservative bound on their radius
of R < 0.9 x 10^-4 mm, for three extra dimensions it is R < 1.9 x 10^-7 mm.
