Catalysis of the reaction: UDP-N-acetyl-D-glucosamine + N-acetyl-D-glucosaminyl-diphosphodolichol = UDP + N,N''-diacetylchitobiosyldiphosphodolichol.
UDP-N-acetyl-D-glucosamine:N-acetyl-D-glucosaminyl-diphosphodolichol N-acetyl-D-glucosaminyltransferase activity