The Beaufort Gyre, located north of Alaska and Canada, is a key circulation system of the Arctic Ocean. Changes in its depth and circulation influence the evolution of the Arctic sea ice cover, the North Atlantic circulation, and the global climate. The gyre is driven by persistent, ice‐mediated winds, accumulating surface freshwater toward the center, deepening the gyre, and spinning up its currents. We describe a mechanism, dubbed here the ice‐ocean governor, in which the interaction of surface currents with the ice regulates the depth of the Beaufort Gyre: The spinning up of the gyre reduces the relative speed between the ocean and the ice, and hence the freshwater accumulation. This competes with, and we argue is more important than, the release of freshwater by flow instability, which moves water from the center toward the periphery. In the current climate the depth and speed of the Beaufort Gyre are mainly set by the governor, but this may change in a warming world where reduced ice cover will render the ice‐ocean governor less effective. The resulting deeper, swifter gyre will likely exhibit more variability in its freshwater storage and flow speeds.