Process for the continuous synthesis of a modified diene elastomer comprising at least one silicon atom directly connected to the elastomer chain, comprising: a) a stage of polymerization by means of n reactors r1 to rn which are equipped with an internal stirring system, which are assumed perfectly stirred and which are arranged in series, n varying from 2 to 15, the reactor r1 being fed by an input solution comprising a solvent, one or more monomer(s), an anionic polymerization initiator chosen from lithium amides and a polar agent, the conversion by weight C 1 in the first reactor being less than 70%, where C 1 = P 1 /W 1 where P 1 is the weight of polymer formed at the outlet of the reactor r1, where W 1 is the amount by weight of monomer(s) introduced into the reactor r1, the total conversion by weight C n at the outlet of the reactor rn being greater than or equal to 70%, where: C n = P n ∑ 1 n W i where P n is the weight of polymer formed at the outlet of the reactor rn, and W i is the amount by weight of monomer(s) introduced into the reactor ri, i varying from 1 to n, b) a stage of modification of the living diene elastomer obtained in the preceding stage by means of a functionalization, coupling or star-branching agent, which makes it possible to introduce, into the elastomer, a functional group comprising at least one silicon atom directly connected to the elastomer chain, the functionalization, coupling or star-branching agent which makes it possible to introduce, into the elastomer, a functional group comprising at least one silicon atom directly connected to the elastomer chain, is at least one compound corresponding to the formula: SiR4 p (OR') 4-p-q (R3-X) q, (Formula II) in which: - R3 is a saturated or unsaturated, cyclic or non-cyclic, divalent C 1 -C 18 aliphatic hydrocarbon radical or divalent C 6 -C 18 aromatic hydrocarbon radical, preferably a linear or branched, divalent aliphatic hydrocarbon radical, more preferably a linear divalent aliphatic hydrocarbon radical and more preferably still the linear C 2 or C 3 hydrocarbon radical; - X is a hydrogen atom or a functional group capable of interacting with a reinforcing filler comprising at least one heteroatom chosen from N, S, O or P; - the R' radicals, which are substituted or unsubstituted and identical or different, represent a C 1 -C 10, indeed even C 1 -C 8, alkyl group, preferably a C 1 -C 4 alkyl group, more preferably methyl and ethyl; - the R4 radicals, which are substituted or unsubstituted and identical or different, represent a C 1 -C 10 alkyl group, indeed even a C 1 -C 8 alkyl group; - p represents an integer with a value ranging from 0 to 2 and q represents an integer with the value 0 or 1, with the proviso that p + q ≤ 2. a) a stage of polymerization by means of n reactors r1 to rn which are equipped with an internal stirring system, which are assumed perfectly stirred and which are arranged in series, n varying from 2 to 15, the reactor r1 being fed by an input solution comprising a solvent, one or more monomer(s), an anionic polymerization initiator chosen from lithium amides and a polar agent, the conversion by weight C 1 in the first reactor being less than 70%, where C 1 = P 1 /W 1 where P 1 is the weight of polymer formed at the outlet of the reactor r1, where W 1 is the amount by weight of monomer(s) introduced into the reactor r1, the total conversion by weight C n at the outlet of the reactor rn being greater than or equal to 70%, where: C n = P n ∑ 1 n W i where P n is the weight of polymer formed at the outlet of the reactor rn, and W i is the amount by weight of monomer(s) introduced into the reactor ri, i varying from 1 to n, the conversion by weight C 1 in the first reactor being less than 70%, where C 1 = P 1 /W 1 where P 1 is the weight of polymer formed at the outlet of the reactor r1, where W 1 is the amount by weight of monomer(s) introduced into the reactor r1, the total conversion by weight C n at the outlet of the reactor rn being greater than or equal to 70%, where: C n = P n ∑ 1 n W i where P n is the weight of polymer formed at the outlet of the reactor rn, and W i is the amount by weight of monomer(s) introduced into the reactor ri, i varying from 1 to n, b) a stage of modification of the living diene elastomer obtained in the preceding stage by means of a functionalization, coupling or star-branching agent, which makes it possible to introduce, into the elastomer, a functional group comprising at least one silicon atom directly connected to the elastomer chain, - R3 is a saturated or unsaturated, cyclic or non-cyclic, divalent C 1 -C 18 aliphatic hydrocarbon radical or divalent C 6 -C 18 aromatic hydrocarbon radical, preferably a linear or branched, divalent aliphatic hydrocarbon radical, more preferably a linear divalent aliphatic hydrocarbon radical and more preferably still the linear C 2 or C 3 hydrocarbon radical; - X is a hydrogen atom or a functional group capable of interacting with a reinforcing filler comprising at least one heteroatom chosen from N, S, O or P; - the R' radicals, which are substituted or unsubstituted and identical or different, represent a C 1 -C 10, indeed even C 1 -C 8, alkyl group, preferably a C 1 -C 4 alkyl group, more preferably methyl and ethyl; - the R4 radicals, which are substituted or unsubstituted and identical or different, represent a C 1 -C 10 alkyl group, indeed even a C 1 -C 8 alkyl group; - p represents an integer with a value ranging from 0 to 2 and q represents an integer with the value 0 or 1, with the proviso that p + q ≤ 2.