The first thermal conductivity measurements of liquid oxygen at temperatures below 80 K, and at pressures up to 1 MPa are reported. The measurements were conducted in a horizontal, guarded, flat plate calorimeter, and are based on the steady-state approach. The cooling power to the calorimeter is provided by a Gifford-McMahon type cryocooler. The temperature difference between the plates of the calorimeter and the absolute temperature are measured using calibrated platinum thermometers. One dimensional heat transfer between the hot and cold plates of the calorimeter is achieved by placing two thermal guards, instrumented with heaters and platinum thermometers, around the hot plate. Although used exclusively for measurements with oxygen, this apparatus is capable of measuring the thermal conductivity of other fluids including liquid hydrogen down to about 15 K.