Mitigation, in the sense of slowing the rate of CO2 emission, can reduce the rate of rise of CO2 and global warming and limit peak CO2 concentration and global warming. However, it will have little effect on the near steady state CO2 concentration and corresponding global warming achieved on a millennial timescale once emissions are negligible and the added carbon has been distributed between the ocean, atmosphere and land. Using a simple model with mid-range climate sensitivity, we estimate that the maximum rate of global warming can be limited to <0.2degreesC/decade by limiting the rate of increase in fossil fuel emissions this century to <0.03 GtC/yr/yr. However, regardless of the emissions pathway, if the known fossil fuel resource of similar to4000 GtC is emitted, CO2 will reach similar to1000 ppmv and the Earth will be warmed by >5degreesC by the end of the millennium. The more carbon emitted, the greater the fraction that remains in the atmosphere, because of positive feedbacks in the carbon cycle-climate system. Less carbon must be emitted to lessen eventual warming. Early consideration should therefore be given to leaving a fraction of fossil carbon unused, and/or to carbon capture and storage.