The Sun has gotten 25% brighter since it formed. Concurrent with this solar brightening, there is evidence for dramatic changes in the composition of Earth’s atmospheric greenhouse. The climatic response to these changes must underpin our understanding of Earth’s physical climate system. However, there has been vigorous debate about the climatic response, as constrained by the oxygen isotopic composition (18O) of marine sedimentary rocks. This ~4-billion-year isotopic record has yielded two contrasting end-member interpretations. In one, 18O of seawater has remained approximately constant, and the temperature of Earth’s surface has decreased from an early 70-80°C to the present global average of ~15°C. In the other, the global average temperature has been approximately constant and 18O of seawater has increased by 10-15 permil. These two end-member scenarios have profoundly different implications for Earth’s climate through geologic time, the mechanisms that regulate it, and the effects on a wide range of Earth-system properties. To address this gap, we developed the 18O of marine iron oxides, which strongly suggests an increase in seawater 18O over Earth history rather than high early temperatures. Using this new record and existing isotopic data, we show that Earth’s temperature has largely been regulated at a global average of ~10-20°C, with implications for long-timescale climate-stabilizing feedbacks. In a spectacular exception to this rule of climate regulation, some of our samples come from a ~50-million-year-long global glaciation, the Sturtian “snowball Earth”. We use these samples to reveal that during this interval a staggering 15-30% of Earth’s water was locked in ice. Despite the frozen surface conditions, our results also require substantial evaporative fluxes from regions of open water and suggest that at least half of the ice came from snowfall in an active hydrological cycle.