NOVELTY - Making a bio-diesel and a bio-naphtha from a complex mixture of natural occurring fats and oils comprises optionally subjecting the complex mixture to a refining treatment, subjecting the complex mixture or refined oils to a hydrolysis step, subjecting the mixture of free fatty acids to a fractionation step to obtain a liquid or substantially liquid free fatty acids part (phase L) and a solid or substantially solid free fatty acids part (phase S), transforming the phase L into alkyl-esters as bio-diesel, and transforming the phase S into linear paraffins as the bio-naphtha.    USE - The process is useful for making a bio-diesel and a bio-naphtha, which is useful as a direct feedstock of a steamcracker, or as blended with at least a conventional feedstock comprising liquefied petroleum gas, naphtha and gasoil, to obtain cracked products including bio-ethylene, bio-propylene, bio-butadiene, bio-isoprene, bio-(di)cyclopentadiene and bio-piperylenes, bio-benzene, bio-toluene, bio-xylene and bio-gasoline (all claimed), where the bio-diesel is useful as an alternative fuel for diesel engines.    ADVANTAGE - The process is capable of producing the bio-diesel and the bio-naphtha with high quality.    DETAILED DESCRIPTION - Making a bio-diesel and a bio-naphtha from a complex mixture of natural occurring fats and oils comprises optionally subjecting the complex mixture to a refining treatment for removing the major part of the non-triglyceride and non-fatty acid components for obtaining refined oils, subjecting the complex mixture or refined oils to a hydrolysis step for obtaining glycerol and a mixture of free fatty acids, subjecting the mixture of free fatty acids to a fractionation step by fractional crystallization for obtaining a liquid or substantially liquid free fatty acids part (phase L) and a solid or substantially solid free fatty acids part (phase S), transforming the phase L into alkyl-esters as bio-diesel by an esterification, and transforming the phase S into linear or substantially linear paraffins as the bio-naphtha by hydrodeoxygenation or decarboxylation of the free fatty acids, where the phase S is obtained fatty acids soaps that are transformed into linear or substantially linear paraffins as the bio-naphtha by decarboxylation of the soaps. An INDEPENDENT CLAIM is included for the use of the bio-naphtha as obtained in the process, as a direct feedstock of a steamcracker, or as blended with at least a conventional feedstock comprising liquefied petroleum gas, naphtha and gasoil, to obtain cracked products including bio-ethylene, bio-propylene, bio-butadiene, bio-isoprene, bio-(di)cyclopentadiene and bio-piperylenes, bio-benzene, bio-toluene, bio-xylene and bio-gasoline.