Catalysis of the reaction: D-galactosyl-(1->4)-beta-D-glucosyl-R + UDP-D-galactose = D-galactosyl-(1->3)-beta-D-galactosyl-(1->4)-beta-D-glucosyl-R + H+ + UDP.
uridine diphosphogalactose-lactosylceramide beta1->3-galactosyltransferase activity