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.
UDPgalactose:D-galactosyl-1,4-beta-D-glucosyl-R beta-1,3-galactosyltransferase activity