An apparatus was developed for the measurement of thermal conductivity under steady conditions. The apparatus uses a 10 nm thick gold film heater that is coated on a polyester sheet. The important advantages of this heater are spatial uniformity of the heat generation and a small thickness. The apparatus is compact and has only a small edge loss of heat. The apparatus was used to measure the thermal conductivity of silica aerogel. The heat transfer modes in silica aerogel include gas and solid conduction and thermal radiation. The experiments provide results for the total heat transfer and hence, total (apparent) thermal conductivity. The separate solid and radiative contributions were obtained from analyses of the energy transfer. The solid conductivity of silica aerogel Varies with temperature more slowly than the radiative conductivity does. The Rosseland absorption coefficient of silica aerogel decreases with temperature. Assuming one-dimensional heat transfer, the uncertainty of the results for the thermal conductivity is about 5.5%. A three-dimensional analysis is used to evaluate the validity of the assumption of one-dimensional heal transfer in the apparatus. The results shows that the assumption of one-dimensional heat transfer in the apparatus results in a reduction of the thermal conductivity of 1.7% at 20 degrees C and a 7.3% reduction at 90 degrees C.