An instrument capable of measuring the polarization characteristics of weakly polarized, cold dust at millimeter and submillimeter wavelengths is presented in detail. The operation and calibration of this polarimeter at a wavelength of 1300 microns, configured for the NRAO 12-meter telescope, are discussed. Deep observations of Jupiter using this instrument revealed a main-beam instrumental polarization at, or below, the 0.2 percent level. Lunar limb observations revealed a sidelobe polarization sensitivity, in the range 0.25 percent - 1.0 percent. Further, through these efforts the nonthermal polarized flux from Jupiter at a level of about 0.04 percent of the thermal flux has been detected. Astronomical polarization measurements to 0.03 percent are possible, limited by the uncertainties in the instrumental polarization. This instrument has been primarily employed to measure and map magnetic-field directions in the very optically opaque cores of massive molecular clouds.