The hyperthermostable beta-glycosidases from the Archaea Sulfolobus solfataricus (SsbetaGly) and Pyrococcus furiosus (CelB) hydrolyse beta-glycosides of D-glucose or D-galactose with relaxed specificities pertaining to the nature of the leaving group and the glycosidic linkage. To determine how specificity is manifested under conditions of kinetically controlled transgalactosylation, the major transfer products formed during the hydrolysis of lactose by these enzymes have been identified, and their appearance and degradation have been determined in dependence of the degree of substrate conversion. CelB and SsbetaGly show a marked preference for making new beta(1-->3) and beta(1-->6) glycosidic bonds by intermolecular as well as intramolecular transfer reactions. The intramolecular galactosyl transfer of CelB, relative to glycosidic-bond cleavage and release of glucose, is about 2.2 times that of SsbetaGly and yields beta-D-Galp-(1-->6)-D-Glc and beta-D-Galp-(1-->3)-D-Glc in a molar ratio of approximately 1 : 2. The partitioning of galactosylated SsbetaGly between reaction with sugars [kNu (M-1. s-1)] and reaction with water [kwater (s-1)] is about twice that of CelB. It gives a mixture of linear beta-D-glycosides, chiefly trisaccharides at early reaction times, in which the prevailing new glycosidic bonds are beta(1-->6) and beta(1-->3) for the reactions catalysed by SsbetaGly and CelB, respectively. The accumulation of beta-D-Galp-(1-->6)-D-Glc at the end of lactose hydrolysis reflects a 3-10-fold specificity of both enzymes for the hydrolysis of beta(1-->3) over beta(1-->6) linked glucosides. Galactosyl transfer from SsbetaGly or CelB to D-glucose occurs with partitioning ratios, kNu/kwater, which are seven and > 170 times those for the reactions of the galactosylated enzymes with 1-propanol and 2-propanol, respectively. Therefore, the binding interactions with nucleophiles contribute chiefly to formation of new beta-glycosides during lactose conversion. Likewise, noncovalent interactions with the glucose leaving group govern the catalytic efficiencies for the hydrolysis of lactose by both enzymes. They are almost fully expressed in the rate-limiting first-order rate constant for the galactosyl transfer from the substrate to the enzyme and lead to a positive deviation by approximately 2.5 log10 units from structure-reactivity correlations based on the pKa of the leaving group.