Beta-1,3-galactosyltransferase that transfers galactose from UDP-galactose to substrates with a terminal beta-linked galactose residue. Has a preference for galactose-beta-1,4-xylose that is found in the linker region of glycosaminoglycans, such as heparan sulfate and chondroitin sulfate. Has no activity towards substrates with terminal glucosamine or galactosamine residues. 3-O-(beta-D-galactosyl-(1->4)-beta-D-xylosyl)-L-seryl-[protein] + UDP-alpha-D-galactose = 3-O-(beta-D-galactosyl-(1->3)-beta-D-galactosyl-(1->4)-beta-D-xylosyl)-L-seryl-[protein] + H(+) + UDP Glycan metabolism; chondroitin sulfate biosynthesis. Glycan metabolism; heparan sulfate biosynthesis. Belongs to the glycosyltransferase 31 family. b3GalT6