This method is for producing a pure bispecific antibody starting from modifying the amino acid sequences of the two monoclonal antibodies to be combined. A mild reduction condition for “splitting the monoclonal antibodies in half” cleaves only the interchain disulfide bonds between two heavy chains, and maintains the cognate light chain-to-heavy chain association or the intact antigen-binding domain (i.e., Fab). The sequence modifications involve the “knobs-into-holes” mutations in the Fc fragment’s CH3 dimerization interface for enabling the two different heavy chains to dimerize or the two different Fab’s to recombine (i.e., Fab-arm exchange) in the formation of bispecific antibody. In addition to high yield, the pure bispecific antibody can be produced through a fast two-step affinity chromatographic method.