GRB 040403 is one of the faintest gamma-ray bursts for which a rapid and accurate localization has been obtained. Here we report on the gamma-ray properties of this burst, based on observations with the IBIS instrument aboard INTEGRAL, and the results of searches for its optical afterglow. The steep spectrum (power law photon index = 1.9 in the 20-200 keV range) implies that GRB 040403 is most likely an X-ray rich burst. Our optical limit of R > 24.2 at 16.5 hours after the burst, indicates a rather faint afterglow, similar to those seen in other relatively soft and faint bursts.