Method for producing block copolymers for tyre tread compositions, and corresponding copolymers

A process for the preparation of a functionalized, coupled or starred block copolymer which is usable in a sulphur-cross-linkable rubber composition of reduced hysteresis in the cross-linked state, one at least of said blocks consisting of a polyisoprene and at least one other block consisting of a diene elastomer other than a polyisoprene the molar ratio of units originating from one or more conjugated dienes of which is greater than 15%. The process includes the steps of : (i) copolymerization of one or more monomers comprising a conjugated diene other than isoprene using a catalytic system comprising a hydrocarbon solvent, a compound A of a metal of group IIIA, a compound B of an alkaline-earth metal and a polymer initiator C comprising a C-Li bond which is formed of a monolithiated polyisoprene intended to form the polyisoprene, said polyisoprene block(s) having a number-average molecular weight Mn1 of between 2, 500 and 20, 000 g/mol, so that said block(s) other than a polyisoprene has/have a content of trans-1, 4 linkages equal to or greater than 70% and a number-average molecular weight Mn2 of between 65, 000 and 350, 000 g/mol, and (ii) addition to the product of copolymerization of a functionalizing, coupling or starring agent comprising one, two or at least three epoxy groups respectively, so that said block other than a polyisoprene is functionalized, coupled or starred.