The residual fraction of anthropogenic CO₂ emissions which has not been captured by carbon sinks and remains in the atmosphere, is estimated by two independent experimental methods which support each other: the ¹³C/¹²C ratio and the temperature-independent fraction of d(CO₂)/dt on a yearly scale after subtraction of annual fluctuations the amplitude ratio of which reaches a factor as large as 7. The anthropogenic fraction is then used to evaluate the additional warming by analysis of its spectral contribution to the outgoing long-wavelength radiation (OLR) measured by infrared spectrometers embarked in satellites looking down. The anthropogenic CO₂ additional warming extrapolated in 2100 is found lower than 0.1°C in the absence of feedbacks. The global temperature data are fitted with an oscillation of period 60 years added to a linear contribution. The data which support the 60-year cycle are summarized, in particular sea surface temperatures and sea level rise measured either by tide gauge or by satellite altimetry. The tiny anthropogenic warming appears consistent with the absence of any detectable change of slope of the 130-year-long linear contribution to the temperature data before and after the onset of large CO₂ emissions.