(Abridged) This paper predicts self-consistent faint galaxy counts from the UV to the submm wavelength range. The STARDUST spectral energy distributions described in Devriendt et al. (1999) are embedded within the explicit cosmological framework of a simple semi-analytic model of galaxy formation and evolution. We build a class of models which capture the luminosity budget of the universe through faint galaxy counts and redshift distributions in the whole wavelength range spanned by our spectra. In contrast with a rather stable behaviour in the optical and even in the far-IR, the submm counts are dramatically sensitive to variations in the cosmological parameters and changes in the star formation history. Faint submm counts are more easily accommodated within an open universe with a low value of $Omega_0$, or a flat universe with a non-zero cosmological constant. This study illustrates the implementation of multi-wavelength spectra into a semi-analytic model. In spite of its simplicity, it already provides fair fits of the current data of faint counts, and a physically motivated way of interpolating and extrapolating these data to other wavelengths and fainter flux levels.
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