Butadiene polymers containing 70% or more of 1,2-structure and a relatively low melting point are produced by polymerizing 1,3-butadiene in the presence of a catalyst which has been prepared by admixing (A) an organic solvent solution containing 1,3-butadiene, a cobalt compound and an organoaluminium compound; (B) an amide compound of the formula (2) or (3): ##STR1## wherein R.sub.1, R.sub.2 and R.sub.3 are respectively an H atom, aliphatic hydrocarbon radical of 1 to 7 carbon atoms or aromatic hydrocarbon radical of 6 or 7 carbon atoms, R.sub.3 is H or an aliphatic hydrocarbon radical of 1 to 3 carbon atoms and n is 2 to 5, and; (C) carbon disulfide.