Beta-1,3-galactosyltransferase that transfers galactose from UDP-galactose to substrates with a terminal beta-N-acetylglucosamine (beta-GlcNAc) residue. Can also utilize substrates with a terminal galactose residue, albeit with lower efficiency. Involved in the biosynthesis of the carbohydrate moieties of glycolipids and glycoproteins. Inactive towards substrates with terminal alpha-N- acetylglucosamine (alpha-GlcNAc) or alpha-N-acetylgalactosamine (alpha- GalNAc) residues. {ECO:0000269|PubMed:9417100, ECO:0000269|PubMed:9582303}. This is the function of B3GALT2 (beta-1,3-galactosyltransferase 2, ENSG00000162630).