The low-temperature properties of superconducting metallic glasses are governed by atomic tunnelling states. The heat transport well below the transition into the superconducting state is limited by the resonant interaction of phonons with the tunnelling systems. So far, measurements of the thermal conductivity have been performed on thin amorphous films down to about 100 mK and on bulk metallic glasses to about 1 K. Using a novel non-contact method, we have investigated for the first time the thermal transport of a superconducting bulk metallic glass Zr_52.5Ti₅Cu_17.9Ni_14.6Al₁₀ down to about 6 mK, testing the prediction of the tunnelling model and searching for a possible influence of nuclear moments on the heat flow. The observed temperature dependence of the thermal conductivity is in reasonable agreement with the prediction of the tunnelling model.