The Mid-Infrared Instrument (MIRI) is one of four instruments on the James Webb Space Telescope observatory, scheduled for launch in 2013. It must be cooled to about 7 K and is supported within the telescope’s 40 K instrument module by a hexapod of carbon fibre reinforced plastic (CFRP) tubing. This article describes the measurement of cryogenic thermal conductivity of the candidate CFRP. Measured thermal conductivities were about 0.05 W/m K at a mean temperature of 10 K increasing to about 0.20 W/m K at a mean temperature of 40 K.