An important outstanding issue in the study of cuprate high temperature superconductors is the physics that controls the disappearance of the superconductivity on the overdoped side of the phase diagram. It is still not clear whether existing observations are dominated by the effects of disorder. To address this, we have performed a careful study of the superconducting anomalies in single crystals of Tl2Ba2CuO6, which have the longest resistively-determined mean free paths of any overdoped cuprates. The experiments required measurement of the electronic heat capacity of microgram single crystals, with transition temperatures tuned from 50 K down to 15 K by annealing. The results and their implications will be discussed.