We have measured the thermal conductivity of an equimolar mixture of methane and ethane in the vicinity of the critical point. The new experimental data confirm that the thermal conductivity of a mixture does not diverge at the critical point but crosses over to a finite limiting behavior at the critical point. A quantitative representation of the thermal-conductivity data has been obtained in terms of a recently developed extension of the mode-coupling theory that incorporates the crossover between the behavior of the thermal conductivity close to and far away from the critical point. The same theory enables us also to predict other transport properties of the mixture in the vicinity of the critical point.