  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~$\pm0.3$~N/clamp. The thermal conductance of a
single sapphire-on-copper clamp was found to be
660$^{+360}_{-210}$~$(T/K)^{3.1}$~[nW/K]. For a sapphire-on-silicon clamp the
conductance was found to be 380$^{+190}_{-120}$~$(T/K)^{2.8}$~[nW/K]. The
conductance measured is consistent with thermal boundary resistance.
