The 40 ultraluminous galaxies in the IRAS Bright Galaxy Sample of sources stronger than S = 5.24 Jy at lambda = 60 microns were mapped with approximately 0.25 arcsec resolution at 8.44 GHz. Twenty-five contain diffuse radio sources obeying the FIR-radio correlation; these are almost certainly starburst galaxies. Fourteen other galaxies have nearly blackbody FIR spectra with color temperatures between 60 and 80 K so their (unmeasured) FIR angular sizes must exceed approximately 0.25 arcsec, yet they contain compact (but usually resolved) radio sources smaller than this limit. The unique radio and FIR properties of these galaxies can be modeled by ultraluminous nuclear starbursts so dense that they 67 are optically thick to free-free absorption at about 1.49 GHz and dust absorption at about 25 microns. Only one galaxy (UGC 08058 = Mrk 231) is a dominated by a variable radio source too compact to be an ultraluminous starburst; it must be powered by a 'monster'.