We report the first counts of faint submillimetre galaxies (SMG) in the 870-um band derived from arcsecond resolution observations with the Atacama Large Millimeter Array (ALMA). We have used ALMA to map a sample of 122 870-um-selected submillimetre sources drawn from the (0.5x0.5)deg^2 LABOCA Extended Chandra Deep Field South Submillimetre Survey (LESS). These ALMA maps have an average depth of sigma(870um)~0.4mJy, some ~3x deeper than the original LABOCA survey and critically the angular resolution is more than an order of magnitude higher, FWHM of ~1.5 compared to ~19 for the LABOCA discovery map. This combination of sensitivity and resolution allows us to precisely pin-point the SMGs contributing to the submillimetre sources from the LABOCA map, free from the effects of confusion. We show that our ALMA-derived SMG counts broadly agree with the submillimetre source counts from previous, lower-resolution single-dish surveys, demonstrating that the bulk of the submillimetre sources are not caused by blending of unresolved SMGs. The difficulty which well-constrained theoretical models have in reproducing the high-surface densities of SMGs, thus remains. However, our observations do show that all of the very brightest sources in the LESS sample, S(870um)>12mJy, comprise emission from multiple, fainter SMGs, each with 870-um fluxes of <9mJy. This implies a natural limit to the star-formation rate in SMGs of <10^3 M_Sun/yr, which in turn suggests that the space densities of z>1 galaxies with gas masses in excess of ~5x10^10 M_Sun is <10^-5 Mpc^-3. We also discuss the influence of this blending on the identification and characterisation of the SMG counterparts to these bright submillimetre sources and suggest that it may be responsible for previous claims that they lie at higher redshifts than fainter SMGs.
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