The invention relates to a method for preparing a polymer conjugate, in which a water-soluble polymer employed, the polymer having a linear or branched water-soluble polymer backbone and, at one end of the backbone, a branched moiety linked with two free reactive groups capable of covalent bonding to another molecule. Most specificly, it relates to an activated, substantially water soluble poly(ethylene glycol) having a linear or branched poly(ethylene glycol) backbone and at least one terminus linked to the backbone through a hydrolytically stable linkage, wherein the terminus is branched and has proximal reactive groups is reacted with active moieties in a biologically active agent such as a protein or peptide thus forming conjugates between the activated poly(ethylene glycol) and the biologically active agent.