We have used the Submillimeter Array to image a flux-limited sample of seven submillimeter galaxies, selected by the AzTEC camera on the JCMT at 1.1 mm, in the COSMOS field at 890 μ m with ~2'' resolution. All of the sources—two radio-bright and five radio-dim—are detected as single point sources at high significance (>6 σ), with positions accurate to ~0.2'' that enable counterpart identification at other wavelengths observed with similarly high angular resolution. All seven have IRAC counterparts, but only two have secure counterparts in deep HST ACS imaging. As compared to the two radio-bright sources in the sample, and those in previous studies, the five radio-dim sources in the sample (1) have systematically higher submillimeter-to-radio flux ratios, (2) have lower IRAC 3.6-8.0 μ m fluxes, and (3) are not detected at 24 μ m . These properties, combined with size constraints at 890 μ m (θ 1.2''), suggest that the radio-dim submillimeter galaxies represent a population of very dusty starbursts, with physical scales similar to local ultraluminous infrared galaxies, with an average redshift higher than radio-bright sources.