Linear polarization maps of the Carina Nebula were obtained at 250, 350, and 500 μm during the 2012 flight of the Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry (BLASTPol). These measurements are combined with Planck 850 μm data in order to produce a submillimeter spectrum of the polarization fraction of the dust emission, averaged over the cloud. This spectrum is flat to within ±15% (relative to the 350 μm polarization fraction). In particular, there is no evidence for a pronounced minimum of the spectrum near 350 μm, as suggested by previous ground-based measurements of other molecular clouds. This result of a flat polarization spectrum in Carina is consistent with recently published BLASTPol measurements of the Vela C molecular cloud and also agrees with a published model for an externally illuminated, dense molecular cloud by Bethell and collaborators. The shape of the spectrum in Carina does not show any dependence on the radiative environment of the dust, as quantified by the Planck-derived dust temperature or dust optical depth at 353 GHz.