NOVELTY - Recovering (M1) lower alkanol from a crude biodiesel containing alkyl esters of fatty acids, lower alkanol, and a catalytically amount of base catalyst, involves subjecting the crude biodiesel that contains water (less than 0.5 mass.%) to fast, vapor fractionation conditions to provide a lower boiling fraction containing lower alkanol.    USE - For recovering lower alkanol such as methanol, ethanol and isopropanol, from crude biodiesel containing alkyl esters of fatty acids, lower alkanol, and base catalyst; and for making biodiesel (claimed).    ADVANTAGE - The process is capable of recovering lower alkanol from crude biodiesel in an economically attractive manner with the recovered lower alkanol having suitable purity to be recycled for transesterification, thus minimizing the loss of unreacted lower alkanol.    DETAILED DESCRIPTION - INDEPENDENT CLAIMS are included for the following:    (1) making (M2) biodiesel involving either: a) contacting a glyceride-containing feed and lower alkanol under transesterification conditions comprising the presence of a base catalyst, where the molar ratio of lower alkanol to glyceride is at least 3.15:1 to provide a crude biodiesel containing alkyl esters of fatty acids, glycerin, lower alkanol, base catalyst, and water (less than 0.1 mass.%), where contacting being for a time sufficient to convert at least 90 mass.% of the glycerides in the glyceride-containing feed; b) separating by phase separation the crude biodiesel to provide a heavier glycerin-containing layer and a lighter alkyl ester-containing layer, where a portion of the water and a portion of the base catalyst are contained in each of the heavier and lighter layer; c) subjecting the lighter layer while it contains a catalytically amount of base catalyst to vapor fractionation conditions to provide a lower boiling fraction containing lower alkanol and a higher boiling fraction containing alkyl esters and base catalyst; d) recycling at least a portion of the lower boiling fraction to step (a) as a portion of the lower alkanol; and e) contacting the higher boiling fraction with an aqueous acid solution in an amount to at least neutralize the base catalyst; or (f) contacting a glyceride-containing feed and lower alkanol under transesterification conditions comprising the presence of a base catalyst, where the molar ratio of lower alkanol to glyceride is at least about 3.15:1 to provide an intermediate containing alkyl esters of fatty acids, glycerin, lower alkanol, base catalyst, and water (less than 0.1 mass.%), where contacting being for a time to convert at least 90 mass.% of the glycerides in the glyceride-containing feed; (g) separating by phase separation the intermediate to provide a heavier glycerin-containing layer and an intermediate lighter alkyl ester-containing layer, where a portion of the water and a portion of the base catalyst are contained in each of the heavier and lighter layer; and (h) contacting the intermediate and lower alkanol, where the molar ratio of alkanol to glyceride in the intermediate is at least 3.15:1 under transesterification conditions comprising the presence of a base catalyst for a time to convert at least 98 mass.% of the glycerides in the glyceride-containing feed and provide a crude biodiesel product; and    (2) an apparatus for conducting the process (M2).