The Beaufort Gyre is a key circulation system of the Arctic Ocean and its main reservoir of freshwater. Freshwater storage and release affects Arctic sea ice cover, as well as North Atlantic and global climate. We describe a mechanism that is fundamental to the dynamics of the gyre, namely, the ice-ocean stress governor. Wind blows over the ice, and the ice drags the ocean. But as the gyre spins up, currents catch the ice up and turn off the surface stress. This governor sets the basic properties of the gyre, such as its depth, freshwater content, and strength. Analytical and numerical modeling is employed to contrast the equilibration processes in an ice-covered versus ice-free gyre. We argue that as the Arctic warms, reduced sea ice extent and more mobile ice will result in a deeper and faster Beaufort Gyre, accumulating more freshwater that will be released by Ekman upwelling or baroclinic instability.