A composition comprising at least one liquid fatty phase structured by at least one semi-crystalline polymer having a low melting point and an organic structure, wherein said polymer is solid at ambient temperature, has a melting temperature of less than 50 C., and comprises a) a polymer backbone and b) at least one crystallizable organic side chain and/or one crystallizable organic block that forms part of the polymer backbone of said polymer, wherein said polymer has a number-average molecular mass of greater than 2000, and wherein the liquid fatty phase and the polymer form a physiologically acceptable medium. This composition is provided in the form of a stick, for example, which deposits, on keratinous substances such as the lips, a glossy, nonsticky and covering film.