In order to provide secure clamping with a low thermal conductance, some low temperature detectors are held with point contact sapphire ball clamps. While this method is increasingly common, the thermal conductance across this interface has not been well studied. We present a direct measurement of the thermal conductance of such clamps between 200 and 600 mK, with a clamping force of approximately 2.7 ± 0.3 N / clamp . The thermal conductance of a single sapphire-on-copper clamp was found to be 660 - 210 + 360 ( T / K ) 3.1 nW/K. For a sapphire-on-silicon clamp the conductance was found to be 380 - 120 + 190 ( T / K ) 2.8 nW/K. The conductance measured is consistent with thermal boundary resistance.