Process for converting petroleum feedstocks comprising an ebullating-bed hydrocracking stage, a maturation stage and a stage of separating the sediments for the production of fuel oils with a low sediment content

The invention relates to a process for converting a hydrocarbon-containing feedstock containing at least one hydrocarbon fraction having a sulphur content of at least 0.1% by weight, an initial boiling temperature of at least 340° C. and a final boiling temperature of at least 440° C., making it possible to obtain a heavy fraction having a sediment content after ageing of less than or equal to 0.1% by weight, said process comprising the following stages: a) a stage of hydrocracking the feedstock in the presence of hydrogen in at least one reactor containing a supported catalyst in an ebullating bed, b) a stage of separating the effluent obtained at the end of stage a), c) a stage of maturation of the heavy fraction originating from the separation stage b), d) a stage of separating the sediments from the heavy fraction originating from the maturation stage c) to obtain said heavy fraction.