The Early Eocene Climate Optimum (EECO, which occurred about 51 to 53 million years ago), was the warmest interval of the past 65 million years, with mean annual surface air temperature over ten degrees Celsius warmer than during the pre-industrial period. Subsequent global cooling in the middle and late Eocene epoch, especially at high latitudes, eventually led to continental ice sheet development in Antarctica in the early Oligocene epoch (about 33.6 million years ago). However, existing estimates place atmospheric carbon dioxide (CO2) levels during the Eocene at 500-3,000 parts per million, and in the absence of tighter constraints carbon-climate interactions over this interval remain uncertain. Here we use recent analytical and methodological developments to generate a new high-fidelity record of CO2 concentrations using the boron isotope (δ(11)B) composition of well preserved planktonic foraminifera from the Tanzania Drilling Project, revising previous estimates. Although species-level uncertainties make absolute values difficult to constrain, CO2 concentrations during the EECO were around 1,400 parts per million. The relative decline in CO2 concentration through the Eocene is more robustly constrained at about fifty per cent, with a further decline into the Oligocene. Provided the latitudinal dependency of sea surface temperature change for a given climate forcing in the Eocene was similar to that of the late Quaternary period, this CO2 decline was sufficient to drive the well documented high- and low-latitude cooling that occurred through the Eocene. Once the change in global temperature between the pre-industrial period and the Eocene caused by the action of all known slow feedbacks (apart from those associated with the carbon cycle) is removed, both the EECO and the late Eocene exhibit an equilibrium climate sensitivity relative to the pre-industrial period of 2.1 to 4.6 degrees Celsius per CO2 doubling (66 per cent confidence), which is similar to the canonical range (1.5 to 4.5 degrees Celsius), indicating that a large fraction of the warmth of the early Eocene greenhouse was driven by increased CO2 concentrations, and that climate sensitivity was relatively constant throughout this period.