Care and/or make-up cosmetic composition, which is transparent or translucent and/or capable of giving a transparent or translucent deposit, comprising a liquid fatty phase comprising at least one ester oil chosen from esters of monocarboxylic acids with monoalcohols and polyalcohols, 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, sulphonamnide, 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., and the liquid fatty phase and the structuring polymer forming a physiologically acceptable medium.