The rise of atmospheric carbon dioxide (CO2) principally due to the burning of fossil fuels is a key driver of anthropogenic climate change. Mitigation strategies include improved efficiency, using renewable energy, and capture and long-term sequestration of CO2. Most sequestration research considers CO2 injection into deep saline aquifers or depleted hydrocarbon reservoirs. Unconventional suggestions include CO2 storage in the porous volcanic lavas of uppermost oceanic crust. Here we test the feasibility of injecting CO2 into deep-sea basalts and identify sites where CO2 should be both physically and gravitationally trapped. We use global databases to estimate pressure and temperature, hence density of CO2 and seawater at the sediment-basement interface. At previously suggested sites on the Juan de Fuca Plate and in the eastern equatorial Pacific Ocean, CO2 is gravitationally unstable. However, we identify five sediment-covered regions where CO2 is denser than seawater, each sufficient for several centuries of anthropogenic CO2 emissions.