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We have conducted a deep radio survey with the Very Large Array at 1.4 GHz ofa region containing the Hubble Deep Field. This survey overlaps previousobservations at 8.5 GHz allowing us to investigate the radio spectralproperties of microjansky sources to flux densities greater than 40 $\mu$Jy at1.4 GHz and greater than 8 $\mu$Jy at 8.5 GHz. A total of 371 sources have beencatalogued at 1.4 GHz as part of a complete sample within 20\arcmin ~of theHDF. The differential source count for this region is only marginallysub-Euclidean and is given by $n(S) = (8.3 \pm 0.4) S^{-2.4 \pm 0.1}$sr$^{-1}$Jy$^{-1}$. Above about 100 $\mu$Jy the radio source count issystematically lower in the HDF as compared to other fields. We conclude thatthere is clustering in our radio sample on size scales of 1\arcmin - 40\arcmin\.  The 1.4 GHz selected sample shows that the radio spectral indices arepreferentially steep ($\bar{\alpha}_{1.4} = 0.85 $) and the sources aremoderately extended with average angular size $\theta$ = 1.8\arcsec . Opticalidentification with disk-type systems at $z \sim $ 0.5-0.8 suggests thatsynchrotron emission, produced by supernovae remnants, is powering the radioemission in the majority of sources. The 8.5 GHz sample contains primarilymoderately flat spectrum sources ($\bar{\alpha}_{8.5} = 0.35$), with less than15% inverted. We argue that we may be observing an increased fraction ofoptically thin bremsstrahlung over synchrotron radiation in these distantstar-forming galaxies.Comment: 31 pages, 5 tables, 10 figures, to appear in Jan. 2000 Ap

The Nature of Radio Emission from Distant Galaxies: The 1.4 GHz Observations