Composition based on silicone oil structured in rigid form, especially for cosmetic use

A physiologically acceptable composition, especially a cosmetic composition, comprising a liquid fatty phase comprising at least one silicone oil, structured with a polymer with a weight-average molecular mass ranging from 500 to 500 000, comprising at least one moiety comprising: at least one polyorganosiloxane group comprising from 1 to 1 000 organosiloxane units in the chain of the moiety of in the form of a graft, and at least two groups capable of establishing hydrogen interactions, the polymer being solid at room temperature and soluble in the liquid fatty phase at a temperature of from 25 to 250° C., and the liquid fatty phase comprising partially or totally volatile oil(s) with a flash point greater than or equal to 40° C. and greater than the softening point of the polymer, on condition that the composition comprises at least 10% by mass of a non-volatile oil and/or of a pasty or viscous product.