  Direct submm imaging has recently revealed the 850-micron background to be
mostly composed of a population of distant ultraluminous infrared galaxies, but
identifying the optical/NIR counterparts to these sources has proved difficult
due to the poor submm spatial resolution. However, the proportionality of both
cm and submm data to the star formation rate suggests that high resolution
radio continuum maps with subarcsecond positional accuracy can be exploited to
locate submm sources. In this paper we present results from a targeted SCUBA
survey of micro-Jansky radio sources in the flanking fields of the Hubble Deep
Field. Even with relatively shallow 850-micron SCUBA observations (>6 mJy at
3-sigma), we were successful at making submm detections of optical/NIR-faint
(I>24 and K~21-22) radio sources, and our counts closely match the bright
counts from submm surveys. Redshift estimates can be made from the ratio of the
submm flux to the radio flux across the 100 GHz break in the spectral energy
distribution. This millimetric redshift estimation places the bright submm
population at z=1-3 where it forms the high redshift tail of the faint radio
population. The star formation rate density (SFRD) due to ultraluminous
infrared galaxies increases by more than two orders of magnitude from z~0 to
z~1-3. The SFRD at high redshift inferred from our >6 mJy submm observations is
comparable to that observed in the UV/optical. (Abridged)
