NOVELTY - Preparing a fatty acid alkyl ester product involves contacting fat- and oil-containing biological material feedstock with a flowing solvent mixture; extracting fats and oils with the flowing solvent mixture; flowing the extracted fat and oil-containing solution; effecting a transesterification reaction between extracted fat and oil-containing solution and flowing solvent mixture and alcohol yielding reaction products comprising glycerol and fatty acid alkyl esters, and flowing solvent mixture; collecting glycerol product; and collecting fatty acid alkyl ester product.    USE - For preparing a fatty acid alkyl ester product from a fat- and oil-containing biological material feedstock (claimed) (where the fatty acid alkyl ester product is biodiesel fuel product).    ADVANTAGE - The process uses solid biomass as a starting material feedstock and generates fatty acid alkyl esters suitable for use as a biodiesel fuel; includes a continuous, multistep process in which fat and oil-containing biomass is extracted with a critical fluid and an alcohol co-solvent (0-20%) such as methanol, the extract is directed into a heated transesterification reaction zone with or without addition of a transesterification reactant alcohol such as methanol and with or without a solid phase catalyst, the glycerol and fatty acid alkyl ester products are then isolated, the spent biomass is removed, and the critical fluid and any excess alcohol used are recycled; and reduces the cost of biodiesel production by obviating the need for using a separate, noncontiguous step for isolating and purifying fats and oils from biomass that is needed in all other biodiesel preparation processes; and reduces the cost of biodiesel production by reducing the need for removing water from the reactants prior to transesterification that is needed in solution phase catalyst biodiesel preparation processes. The solid oilseed material can be used in the biodiesel fuel preparation process, without the need for prior isolation and purification of fats and oils from the oilseed.    DETAILED DESCRIPTION - Preparing a fatty acid alkyl ester product from a fat- and oil-containing biological material feedstock involves contacting the biological material feedstock with a flowing solvent mixture where the solvent mixture is maintained at supercritical or near-critical pressure and temperature conditions; extracting fats and oils from the biological material feedstock with the flowing solvent mixture; flowing the extracted fat and oil-containing solution, under pressure, into a separate reaction region maintained at a desired temperature; effecting a transesterification reaction between the extracted fat and oil-containing solution and the flowing solvent mixture and an alcohol within the reaction region at supercritical conditions yielding reaction products comprising glycerol and fatty acid alkyl esters, and the flowing solvent mixture; flowing under pressure reaction products and into a separation region; collecting glycerol product; collecting fatty acid alkyl ester product; collecting wastes; and collecting the flowing solvent mixture and any unused reactants.