Measurements at a wavelength of 1.25 mm have been obtained for 17 IRAS galaxies selected on the basis of high far-infrared luminosity. These measurements are used to estimate the lower and upper limits to the mass of cold dust in infrared galaxies. As a lower limit on dust mass, all of the galaxies can be successfully modeled without invoking any dust colder than the dust responsible for the 60 and 100 micron emission that was detected by IRAS. As an upper limit, it is possible that the dust mass in a number of the galaxies may actually be dominated by cold dust. This large difference between the lower and upper limits is due primarily to uncertainty in the long-wavelength absorption efficiency of the astrophysical dust grains.