TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Components: The catalyst used in two hydrocracking comprises or is constituted of cobalt molybdenum and/or nickel molybdenum.    TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Components: The charge treated by FCC further comprises mild hydrocracking effluent having an initial boiling point of greater than 330 degrees C, a portion of vacuum gas oil (VGO) and/or deasphalted oil (DAO), preferably from the hydrocarbon charge for the mild hydrocracking. The charge treated by the FCC further comprising the effluent, portion of VGO and/or DAO, has a sulfur content of less than or equal to 2500 part per million (ppm), preferably less than or equal to 1000 ppm. The content of the hydrocarbon charge in the vegetable oil or animal fat type biological filler is less than or equal to 30 wt.% with respect to the total weight of the hydrocarbon charge. The filler content of biological origin is less than or equal to 15 wt.% with respect to the total weight of the hydrocarbon charge. The petroleum charge is distillates of vacuum distillation, diesel obtained from the conversion process and/or deasphalted residues. The charge treated by the FCC comprises a portion of VGO and/or DAO, preferably from the hydrocarbon charge for the mild hydrocracking, other than the effluent. The vegetable oils are palm oil, soybean oil, rapeseed oil, sunflower oil, linseed oil, rice bran oil, corn oil, olive oil, castor oil, sesame oil, pine oil, peanut oil, palm kernel oil, coconut oil, babasu oil, preferably palm oil or a mixture of at least two of the oil. Preferred Method: The biological filler is mixed with the petroleum charge before its introduction in the mild hydrocracking reactor. The petroleum charge is injected in a first catalytic area of the mild hydrocracking unit and in that the biological filler is injected in a second catalytic area of the mild hydrocracking unit located downstream of the first catalytic area. The process comprises recycling gas treatment before its reinjection, preferably in the pretreatment reactor, where the process is carrying out an additional treatment during which treating carbon monoxide present in the recycling gas and separating the recycling gas before its injection, is in the mild hydrocracking reactor.