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We present the first results of a sub-millimeter survey of distant clusters using the new bolometer array, SCUBA, on the JCMT. We have mapped fields in two clusters, A370 at z=0. 37 and Cl2244-02 at z=0. 33, at 450 and 850-um. The resulting maps cover a total area of about 10 sq arcmin to 1 sigma noise levels less than 14 and 2 mJy/beam at the two wavelengths. These sensitivities are 2-3 orders of magnitude deeper than was possible with the previous generations of bolometers over such areas. We have concentrated on lensing clusters to exploit the amplification of all background sources by the cluster, improving the sensitivity by a factor of 1. 3-2 as compared with a blank-field survey. A cumulative source surface density of (2. 4+/-1. 0) x 10³ per sq degree is found to a 50% completeness limit of ~4 mJy at 850-um. The sub-mm spectral properties of these sources indicate that the majority lie at high redshift, z>>1. The observed surface density is 3 orders of magnitude greater than the expectation of a non-evolving model using the local IRAS 60-um luminosity function. This indicates a substantial increase in the number density of strongly star-forming galaxies in the high-redshift Universe. The magnitude of this increase implies that optical surveys of the star-formation properties of the distant Universe have missed substantial numbers of obscured star-forming galaxies. The deeper sub-mm surveys planned with SCUBA for the near future should detect large numbers of star-forming galaxies at high redshift, and so provide strong constraints on the formation of normal galaxies. abridged
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