Catalyzes the transfer of a sulfate group from 3'- phosphoadenylyl sulfate (PAPS) to the C-3 hydroxyl group of terminal non-reducing beta-galactosyl residues (PubMed:11029462, PubMed:14701868, PubMed:15926885). Acts both on type 1 (Gal-beta-1,3- GlcNAc-R) and type 2 (Gal-beta-1,4-GlcNAc-R) chains with similar efficiency (PubMed:11029462, PubMed:14701868, PubMed:15926885). Displays species-dependent substrate specificity (PubMed:15926885). Participates in galactose sulfation and shows substrate preference for structures such as lactose, lacto-N-tetraose (Lc4), lacto-N-neotetraose (nLc4), the Gal-beta-1,4-GlcNAc-beta unit in mucin core-2, and the cancer antigen Globo H precursor Gal-beta-1,3-GlcNAc-beta-1,3-Gal-R (PubMed:11029462, PubMed:14701868, PubMed:15926885). {ECO:0000269|PubMed:11029462, ECO:0000269|PubMed:14701868, ECO:0000269|PubMed:15926885}. This is the function of GAL3ST2 (galactose-3-O-sulfotransferase 2, ENSG00000154252).