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GRB 040403 is one of the faintest gamma-ray bursts for which a rapid andaccurate localization has been obtained. Here we report on the gamma-rayproperties of this burst, based on observations with the IBIS instrument aboardINTEGRAL, and the results of searches for its optical afterglow. The steepspectrum (power law photon index = 1.9 in the 20-200 keV range) implies thatGRB 040403 is most likely an X-ray rich burst. Our optical limit of R > 24.2 at16.5 hours after the burst, indicates a rather faint afterglow, similar tothose seen in other relatively soft and faint bursts.Comment: 5 pages, 3 figures, accepted for publication in Astronomy and  Astrophysics Letter

GRB 040403: A faint X-ray rich gamma-ray burst discovered by INTEGRAL