Care and/or make-up cosmetic composition comprising: a liquid continuous fatty Phase structured with at least one structuring polymer (homopolymer or copolymer) having a weight-average molecular mass ranging from 500 to 500 000, containing at least one moiety comprising: at least one polyorganosiloxane group consisting of 1 to 1 000 organosiloxane units in the chain of the moiety or in the form of a graft, and at least two groups capable of establishing hydrogen interactions, Chosen from ester, amide, sulphonamide, carbamate, thiocarbamate, urea, urethane, thiourea, oxamido, guanidino and biguanidino groups, and combinations thereof, the polymer being solid at room temperature and soluble in the liquid fatty Phase at a temperature of 25 to 250° C., the Said liquid fatty Phase comprising at least one compound capable of reducing the enthalpy of fusion of the structuring polymer, and then the liquid fatty Phase, the structuring polymer and the compound capable of reducing the enthalpy of fusion of the structuring polymer forming a physiologically acceptable medium.