The process in which the anatomical structures of the umbilical cord are generated and organized. The umbilical cord is an organ or embryonic origin consisting of the 2 umbilical arteries and the one umbilical vein. The umbilical cord connects the cardiovascular system of the fetus to the mother via the placenta. [ GOC:BHF GOC:gr http://www.ncbi.nlm.nih.gov/pubmed/15107403 ]