Geological evidence suggests that marine ice extended to the Equator at least twice during the Neoproterozoic era (about 750 to 635 million years ago), inspiring the Snowball Earth hypothesis that the Earth was globally ice-covered. In a possible Snowball Earth climate, ocean circulation and mixing processes would have set the melting and freezing rates that determine ice thickness, would have influenced the survival of photosynthetic life, and may provide important constraints for the interpretation of geochemical and sedimentological observations. Here we show that in a Snowball Earth, the ocean would have been well mixed and characterized by a dynamic circulation, with vigorous equatorial meridional overturning circulation, zonal equatorial jets, a well developed eddy field, strong coastal upwelling and convective mixing. This is in contrast to the sluggish ocean often expected in a Snowball Earth scenario owing to the insulation of the ocean from atmospheric forcing by the thick ice cover. As a result of vigorous convective mixing, the ocean temperature, salinity and density were either uniform in the vertical direction or weakly stratified in a few locations. Our results are based on a model that couples ice flow and ocean circulation, and is driven by a weak geothermal heat flux under a global ice cover about a kilometre thick. Compared with the modern ocean, the Snowball Earth ocean had far larger vertical mixing rates, and comparable horizontal mixing by ocean eddies. The strong circulation and coastal upwelling resulted in melting rates near continents as much as ten times larger than previously estimated. Although we cannot resolve the debate over the existence of global ice cover, we discuss the implications for the nutrient supply of photosynthetic activity and for banded iron formations. Our insights and constraints on ocean dynamics may help resolve the Snowball Earth controversy when combined with future geochemical and geological observations.