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Current climate change may induce positive carbon cycle feedbacks that amplify anthropogenic warming on time scales of centuries to millennia. Similar feedbacks might have been active during a phase of carbon cycle perturbation and global warming, termed the Paleocene-Eocene Thermal Maximum (PETM, 56 million years ago). The PETM may help constrain these feedbacks and their sensitivity to warming. We present new high-resolution carbon isotope and sea surface temperature data from Ocean Drilling Program Site 959 in the Equatorial Atlantic. With these and existing data from the New Jersey Shelf and Maud Rise, Southern Ocean, we quantify the lead-lag relation between PETM warming and the carbon input that caused the carbon isotope excursion (CIE). We show similar to 2 degrees C of global warming preceded the CIE by millennia, strongly implicating CO2-driven warming triggered a positive carbon cycle feedback. We further compile new and published barium (Ba) records encompassing continental shelf, slope, and deep ocean settings. Based on this compilation, we calculate that average Ba burial rates approximately tripled during the PETM, which may require an additional source of Ba to the ocean. Although the precipitation pathway is not well constrained, dissolved Ba stored in sulfate-depleted pore waters below methane hydrates could represent an additional source. We speculate the most complete explanation for early warming and rise in Ba supply is that hydrate dissociation acted as a positive feedback and caused the CIE. These results imply hydrates are more temperature sensitive than previously considered, and may warrant reconsideration of the political assignment of 2 degrees C warming as a safe future scenario.