The invention concerns a process for converting a hydrocarbon feed, said process comprising the following steps: a) a step of hydrocracking the feed in the presence of hydrogen; b) a step of separating the effluent obtained from step a); c) a step of precipitating sediments, in which the heavy fraction obtained from the separation step b) is brought into contact with a distillate cut at least 20% by weight of which has a boiling point of 100° C. or more for a period of less than 500 minutes, at a temperature in the range 25° C. to 350° C., and at a pressure of less than 20 MPa; d) a step of physical separation of the sediments from the heavy fraction obtained from step c); e) a step of recovering a heavy fraction having a sediment content, measured using the ISO 10307-2 method, of 0.1% by weight or less.