Carbon single-wall nanotubes (SWNTs) grow efficiently from methane on alumina-supported metal catalysts within a fairly narrow temperature window from 680 to 850 degreesC. An abrupt onset in SWNT growth occurs at the low temperature side of the window, and SWNTs produced at these lower temperatures appear to be relatively free of amorphous or nanocrystalline carbon impurities. Raman spectroscopy shows that SWNT yield drops dramatically at the high-temperature side of the window where most previous chemical vapor decomposition (CVD) studies have been performed. The turn-on at the low-temperature side appears to be controlled by the thermodynamics of SWNT growth, while the turn-off at high temperatures is associated with competitive deposition of amorphous and nanocrystalline carbon. The existence of a temperature window for SWNT growth has not been reported elsewhere, and has important general consequences for CVD growth of SWNTs.