A discussion of what is known experimentally about the thermal contact problem at very low temperatures is given. Liquid4He is a particularly promising contact agent. Methods of using liquid4He, thermometry using cerium magnesium nitrate, and solutions to the filling and isolation problems using an open capillary are discussed. The techniques are applied to an apparatus suitable for heat capacity measurements to below 0.02°K and to an apparatus for measurement of superconductive properties at temperatures substantially below 0.01°K. The performance of the apparatus for heat capacity and superconductivity measurements is discussed.