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: copolymerization of one or more monomers comprising a conjugated diene other than isoprene using a catalytic system comprising at least one 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 block, and addition to the product of the copolymerization of a functionalizing, coupling or starring agent comprising an acetoxy group and of formula R.sub.n--Sn--(O--CO--R').sub.4-n, where n is a natural integer of from 0 to 4 and where R and R' are each alkyl, cycloalkyl, aryl or aralkyl groups which may be identical or different, so that said or each block formed of a diene elastomer other than a polyisoprene is functionalized, coupled or starred.