Earth's atmospheric CO2 concentration (ca) for the Phanerozoic Eon is estimated from proxies and geochemical carbon cycle models. Most estimates come with large, sometimes unbounded uncertainty. Here, we calculate tightly constrained estimates of ca using a universal equation for leaf gas exchange, with key variables obtained directly from the carbon isotope composition and stomatal anatomy of fossil leaves. Our new estimates, validated against ice cores and direct measurements of ca, are less than 1000 ppm for most of the Phanerozoic, from the Devonian to the present, coincident with the appearance and global proliferation of forests. Uncertainties, obtained from Monte Carlo simulations, are typically less than for ca estimates from other approaches. These results provide critical new empirical support for the emerging view that large (~2000-3000 ppm), long-term swings in ca do not characterize the post-Devonian and that Earth's long-term climate sensitivity to ca is greater than originally thought.