The enormous amount of basic research into carbon nanotubes has sparked interest in the potential applications of these novel materials. One promising use of carbon nanotubes is as fillers in a composite material to improve mechanical behaviour, electrical transport and thermal transport. For composite materials with high thermal conductivity, the thermal conductance across the nanotube-matrix interface is of particular interest. Here we use picosecond transient absorption to measure the interface thermal conductance (G) of carbon nanotubes suspended in surfactant micelles in water. Classical molecular dynamics simulations of heat transfer from a carbon nanotube to a model hydrocarbon liquid are in agreement with experiment. Our findings indicate that heat transport in a nanotube composite material will be limited by the exceptionally small interface thermal conductance (G approximately 12 MW m(-2) K(-1)) and that the thermal conductivity of the composite will be much lower than the value estimated from the intrinsic thermal conductivity of the nanotubes and their volume fraction.