Climate change confronts central banks with two inflationary challenges: climateflation and greenflation. We investigate their implications for monetary policy by developing and estimating a tractable nonlinear New Keynesian Climate model featuring climate damages and mitigation policies for the global economy. We find that mitigation policies aligned with the Paris Agreement result in higher, more persistent inflation than laissez-faire policies. Central banks can attenuate this inflationary pressure by accounting for the rising natural rate of interest, at the cost of lower GDP during the transition. This short-term trade-off ensures long-term macroeconomic stability resulting from a net-zero emission world.