Polarized emission from aligned interstellar dust is both a crucial tool for studies of magnetism in the interstellar medium and a troublesome contaminant in studies of the polarized cosmic microwave background. In each case, an understanding of the significance of the dust polarization signal requires well-calibrated models that accurately describe dust grains’ physical properties and interactions with their environment. Despite decades of progress in both theory and observation, polarized dust emission models remain largely underconstrained. During its 2012 flight, BLASTPol (the Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry) obtained simultaneous broad-band polarimetric maps at 250, 350, and 500 μm of a several degree-scale region containing several low-AV molecular clouds. Combining these data with polarimetric observations from the Planck 850 μm band, we have produced a submillimeter polarization spectrum for one of these objects for the first time. We find the polarization degree to be largely constant across the four submillimeter bands. This result introduces a new observable with the potential to place strong empirical constraints on polarized dust models of the ISM in a density regime that has not been accessible to previous experiments. Comparing with the work of Draine & Fraisse (2009), our result is inconsistent with two of their four models. In particular, the two models for which all polarization arises from the aligned silicate component yield submillimeter polarization spectra that rise steeply with wavelength, in disagreement with our observations. This line of investigation will continue in the near future, as new experiments like The Next-Generation BLAST Polarimeter (BLAST-TNG) use their enhanced sensitivities to characterize polarized dust emission in even more diffuse environments.