Method of calcining lime or dolomite in a annular shaft kiln having an outer cylinder and an inner cylinder forming an annular space between them, comprising - a supply of limestone or dolomite material to be calcined at the top of the annular space into which it descends, - introducing first combustion fumes into said annular space, at a first level of the kiln, - introducing second combustion fumes into said annular space, at a second level of the kiln lower than the aforementioned first level, - preheating and calcining of the material to be calcined during descent into the annular space, by contact with the first combustion fumes and second combustion fumes which undergo an upward draught, in counter-current to the material to be calcined, and are removed at the top of the kiln in the form of an upper gas stream, - cooling of the calcined material by supplying product cooling air at the bottom of the annular space, - suctioning this product cooling air inside the inner cylinder, at a third level of the kiln lower than the aforementioned second level, forming therein a lower gas stream containing the product cooling air, and - discharging calcined lime or dolomite at the bottom of the kiln, characterised in that it comprises - withdrawing a portion of the upper gas stream, discharged from the kiln, - a formation of a first oxidising mixture by mixing this sampled portion of the upper gas stream with pure dioxygen, - a combustion of a fuel in this first oxidising mixture, to form said second combustion fumes, and - an extraction of the lower gas stream containing the product cooling air from the inner cylinder, followed by its removal from the kiln. - a supply of limestone or dolomite material to be calcined at the top of the annular space into which it descends, - introducing first combustion fumes into said annular space, at a first level of the kiln, - introducing second combustion fumes into said annular space, at a second level of the kiln lower than the aforementioned first level, - preheating and calcining of the material to be calcined during descent into the annular space, by contact with the first combustion fumes and second combustion fumes which undergo an upward draught, in counter-current to the material to be calcined, and are removed at the top of the kiln in the form of an upper gas stream, - cooling of the calcined material by supplying product cooling air at the bottom of the annular space, - suctioning this product cooling air inside the inner cylinder, at a third level of the kiln lower than the aforementioned second level, forming therein a lower gas stream containing the product cooling air, and - discharging calcined lime or dolomite at the bottom of the kiln, characterised in that it comprises - withdrawing a portion of the upper gas stream, discharged from the kiln, - a formation of a first oxidising mixture by mixing this sampled portion of the upper gas stream with pure dioxygen, - a combustion of a fuel in this first oxidising mixture, to form said second combustion fumes, and - an extraction of the lower gas stream containing the product cooling air from the inner cylinder, followed by its removal from the kiln. <EOS>