Single-crystal sapphire is currently in use as the material for output windows in high-power microwave tubes, particularly gyrotrons. These windows are currently being cooled by fluorocarbon fluids at near-room temperatures. There are, however, several advantages in operating the window at very low temperatures: less absorption and consequent heating of the window, greater material strength, improved resistance to crack formation, greater thermal conductivity, and reduced thermal expansion. Operation at cryogenic temperatures is shown to be feasible. The output power, which is currently limited by window constraints, could be increased by an order of magnitude or more.