Context. The Cygnus region hosts a giant molecular-cloud complex which actively forms massive stars. Interactions of cosmic rays with interstellar gas and radiation fields make it shine at y-ray energies. Several gamma-ray pulsars and other energetic sources are seen in this direction. Aims. In this paper we analyse the gamma-ray emission measured by the Fermi Large Area Telescope in the energy range from 100 Me V to 100 Ge V in order to probe the gas and cosmic-ray content over the scale of the whole Cygnus complex. The gamma-ray emission on the scale of the central massive stellar clusters and from individual sources is addressed elsewhere. Methods. The signal from bright pulsars is largely reduced by selecting photons in their off-pulse phase intervals. We compare the diffuse gamma-ray emission with interstellar gas maps derived from radio/mm-wave lines and visual extinction data. and a global model of the region, including other pulsars and gamma-ray sources, is sought. Results. The integral H I emissivity above 100 MeV averaged over the whole Cygnus complex amounts to 12.06 +/- 0.11 (stat.) (+0.15 -0.84) (syst.J] x 10(exp -26) photons /s / sr / H-atom, where the systematic error is dominated by the uncertainty on the H I opacity to calculate its column densities. The integral emissivity and its spectral energy distribution are both consistent within the systematics with LAT measurements in the interstellar space near the solar system. The average X(sub co) N(H2)/W(sub co) ratio is found to be [1.68 +/- 0.05 (stat.) (H I opacity)] x 1020 molecules cm-2 (K km/s /r, consistent with other LAT measurements in the Local Arm. We detect significant gamma-ray emission from dark neutral gas for a mass corresponding to approx 40% of that traced by CO. The total interstellar mass in the Cygnus complex inferred from its gamma-ray emission amounts to 8(+5 -1) x 10(exp 6) Solar M at a distance of 1.4 kpc. Conclusions. Despite the conspicuous star formation activity and large masses of the interstellar clouds, the cosmic-ray population in the Cygnus complex averaged over a few hundred parsecs is similar to that of the local interstellar space
