Recent x-ray observations have discovered that groups and clusters of galaxies contain many more baryons in intergalactic gas than in stars. If high-energy cosmic rays are present there with an average intensity comparable to that observed in our Galaxy, they could produce the extragalactic diffuse gamma radiation.PACS nuInbers: 98.70. Rz, 98.62.Ra, 98.70.Vc Observations with the gamma ray satellite SAS-II have shown [1] that in addition to the galactic diffuse gamma radiation, which varies strongly with direction and can be explained by cosmic ray interactions in the galactic interstellar medium, there appears to be an unaccounted diffuse component which is isotropic at least on a coarse scale and fits well at low photon energies to the extragalactic hard x-ray background radiation.These two features suggest an extragalactic origin of this isotropic component [1]. Its spectrum in the energy range 35 -300 MeV was fitted [2] by a power law with a spectral index -2.35 03 and an intensity Ii, () 100 MeV) = (0.7 -2.3) X 10 s y cm 2 s 'sr ' at energies greater than 100 MeV. Recent analyses [3 -5] of observations with the Energetic Gamma Ray Experiment Telescope (EGRET) on board the Compton Gamma Ray Observatory (CGRO) have yielded similar results. For instance, a detailed analysis [3] of the high-quality data from "phase 1" of the EGRET observations obtained the result that the differential intensity of the GBR in the energy range 50 -10 GeV is well represented by dI =96' 10 E cm s 'sr 'GeV dF with n = 2.11~0.05. Equation (1) yields an intensity I~(F ) 100 MeV) = (1.1~0.05) X 10 s y cm 2 X s ' sr ' at energies greater than 100 MeV. Since cosmic ray interactions in the interstellar gas of our Milky Way (MW) galaxy explain [6,7] most of the gamma ray emission of the MW, it was suggested [8] that cosmic ray interactions in external galaxies may explain the extragalactic diffuse gamma background radiation (GBR). However, it was found that the summed emission from cosmic ray interactions in external galaxies falls short by a large factor ()20) in explaining the observed intensity of the GBR [9]. This can be seen easily:The total emissivity of the MW in high energy gamma rays (E ) 100 MeV) was estimated [6] to be L~()100 MeV) = 1.3 X 10 y s ', while its optical luminosity was estimated [10] to be LMw = (2.30 .6) X 10'OLO. The average luminosity density of the Universe was measured [11]to be pL = (1.83~0.35) X 10 hI. OMpc, where Ho = 100h kms ' Mpc ' is the Hubble constant. If the gamma ray and optical emis-(2) sivities of galaxies are approximately proportional to the galactic mass, then their ratio is approximately universal and equal to that of the MW. The intensity of the extragalactic GBR is then given by p, L,(oZ) I, ()F. ) = 4~HO I Mw It yields I~()100 MeV) = (2.6~0.8) X 10 y X cm s ' sr ', independent of the value of h. This intensity is smaller by a factor 30 -80 than the observed intensity of the GBR [12].Whereas the ratio of gas mass to stellar light for the MW is Ms"/LMw = 4.8 X 109 Mo/2. 3 X 10'o Lo = 0 21Mo/...