Process for the demercurisation of flue gases, in which fumes to be treated are sent to a gas-solid separator, in which a demercurisation reagent is introduced into the fumes to be treated upstream of the gas-solid separator, and in which the demercurisation reagent is dosed by carrying out the following operations at each instant: a) we determine: - an upstream concentration value (Ca) for the mercury concentration in the fumes upstream of the point of introduction of the demercurisation reagent into the fumes to be treated, and - a flow rate value representative of the flow rate of the flue gases to be treated, b) a first dosing factor is calculated from the upstream concentration value (Ca) and a first function (f) which increases as a function of the upstream concentration value, c) a quantity of mercury is calculated by summing, over a predetermined period of time ending at the instant in question, the product of the upstream concentration value (Ca) and the flow rate value, d) from the upstream concentration value (Ca), the occurrence of a mercury peak is monitored and, each time a mercury peak occurs: - the time t0 at which the peak occurs is recorded, together with the value K0 of the first dosing factor at time t0, and - at each instant following the instant t0, a second dosing factor is calculated from the value K0 and a second function (g) which decreases with time and whose decrease with time is all the slower as the quantity of mercury calculated at the instant in question is greater, and e) the calculated value of the first dosing factor is compared with the calculated value of the second dosing factor and the larger (Kf) of these two calculated values is used to dose the demercurisation reagent. in which fumes to be treated are sent to a gas-solid separator, in which a demercurisation reagent is introduced into the fumes to be treated upstream of the gas-solid separator, and in which the demercurisation reagent is dosed by carrying out the following operations at each instant: a) we determine: - an upstream concentration value (Ca) for the mercury concentration in the fumes upstream of the point of introduction of the demercurisation reagent into the fumes to be treated, and - a flow rate value representative of the flow rate of the flue gases to be treated, b) a first dosing factor is calculated from the upstream concentration value (Ca) and a first function (f) which increases as a function of the upstream concentration value, c) a quantity of mercury is calculated by summing, over a predetermined period of time ending at the instant in question, the product of the upstream concentration value (Ca) and the flow rate value, d) from the upstream concentration value (Ca), the occurrence of a mercury peak is monitored and, each time a mercury peak occurs: - the time t0 at which the peak occurs is recorded, together with the value K0 of the first dosing factor at time t0, and - at each instant following the instant t0, a second dosing factor is calculated from the value K0 and a second function (g) which decreases with time and whose decrease with time is all the slower as the quantity of mercury calculated at the instant in question is greater, and e) the calculated value of the first dosing factor is compared with the calculated value of the second dosing factor and the larger (Kf) of these two calculated values is used to dose the demercurisation reagent. a) we determine: - an upstream concentration value (Ca) for the mercury concentration in the fumes upstream of the point of introduction of the demercurisation reagent into the fumes to be treated, and - a flow rate value representative of the flow rate of the flue gases to be treated, - an upstream concentration value (Ca) for the mercury concentration in the fumes upstream of the point of introduction of the demercurisation reagent into the fumes to be treated, and - a flow rate value representative of the flow rate of the flue gases to be treated, b) a first dosing factor is calculated from the upstream concentration value (Ca) and a first function (f) which increases as a function of the upstream concentration value, c) a quantity of mercury is calculated by summing, over a predetermined period of time ending at the instant in question, the product of the upstream concentration value (Ca) and the flow rate value, d) from the upstream concentration value (Ca), the occurrence of a mercury peak is monitored and, each time a mercury peak occurs: - the time t0 at which the peak occurs is recorded, together with the value K0 of the first dosing factor at time t0, and - at each instant following the instant t0, a second dosing factor is calculated from the value K0 and a second function (g) which decreases with time and whose decrease with time is all the slower as the quantity of mercury calculated at the instant in question is greater, and - the time t0 at which the peak occurs is recorded, together with the value K0 of the first dosing factor at time t0, and - at each instant following the instant t0, a second dosing factor is calculated from the value K0 and a second function (g) which decreases with time and whose decrease with time is all the slower as the quantity of mercury calculated at the instant in question is greater, and e) the calculated value of the first dosing factor is compared with the calculated value of the second dosing factor and the larger (Kf) of these two calculated values is used to dose the demercurisation reagent.