For an optimally matched detector of gravitational radiation pulses, the sensitivity depends largely on the quality factor, Q, of the antenna material. Although in general the Q's of many materials tend to increase at low temperatures, their temperature dependence is marked by relative minima that can affect the sensitivity of the detector. The determination of these minima is therefore of great importance in the selection of the useful temperature range in which to operate the detector. The measurements reported here refer in particular to the first quartz antenna developed by the Regina Group. Its resonant frequency is 25.2 kHz and the measurements span the range from 300 K down to 1 K.