A process for the synthesis of nucleic acids comprising at least one cycle of elongation of nucleic acid fragments, known as initial fragments comprising n nucleotides, n ranging from 3 to 10 12, each cycle being subdivided in the following way: a) A phase of enzymatic addition of Xi nucleotides to one end of said fragments, it being possible for X to be between 1 and 5, preferably between 1 and 3, i being the cycle number, making it possible to obtain fragments comprising n + Xi nucleotides, and comprising the following stages: - a first stage of attaching, to a first support, the 5' end of initial nucleic acid fragments or nucleic acid fragments in the course of elongation, including n nucleotides, said first support having an attaching surface allowing a reversible connection with the 5' end of said fragments, - a stage of addition of the reagents necessary for the enzymatic addition, a stage of enzymatic addition of Xi nucleotides to the second end of said nucleic acid fragments, the free end of the nucleotide(s) added to the fragments has a protective chemical group, to prevent multiple addition of Xi nucleotides to the same fragment, - a stage of detaching from said first support said fragments comprising n + Xi nucleotides, a first stage of transfer of said fragments comprising n + Xi nucleotides, by creating a mobile fluid flow, to a second support capable of interacting with the protective group of the 3' end of the fragments comprising n + Xi nucleotides by means of a non-covalent bond, b) a phase of purification of the fragments having a correct sequence comprising n + Xi nucleotides, comprising the following successive stages: - a second stage of attaching, to said second support, said fragments comprising n + Xi nucleotides by their 3' end, - a stage of removal of the fragments which have not been added to and of the fragments which have not been attached to the said second support, - a stage of detaching said fragments comprising n + Xi nucleotides from said second support each cycle being carried out in a reaction medium compatible with an enzymatic addition and an enzymatic amplification, the synthesis process also comprising, at the end of all of the i elongation cycles, a stage of final amplification by a multiplication factor Yf. a) A phase of enzymatic addition of Xi nucleotides to one end of said fragments, it being possible for X to be between 1 and 5, preferably between 1 and 3, i being the cycle number, making it possible to obtain fragments comprising n + Xi nucleotides, and comprising the following stages: - a first stage of attaching, to a first support, the 5' end of initial nucleic acid fragments or nucleic acid fragments in the course of elongation, including n nucleotides, said first support having an attaching surface allowing a reversible connection with the 5' end of said fragments, - a stage of addition of the reagents necessary for the enzymatic addition, a stage of enzymatic addition of Xi nucleotides to the second end of said nucleic acid fragments, the free end of the nucleotide(s) added to the fragments has a protective chemical group, to prevent multiple addition of Xi nucleotides to the same fragment, - a stage of detaching from said first support said fragments comprising n + Xi nucleotides, a first stage of transfer of said fragments comprising n + Xi nucleotides, by creating a mobile fluid flow, to a second support capable of interacting with the protective group of the 3' end of the fragments comprising n + Xi nucleotides by means of a non-covalent bond, - a first stage of attaching, to a first support, the 5' end of initial nucleic acid fragments or nucleic acid fragments in the course of elongation, including n nucleotides, said first support having an attaching surface allowing a reversible connection with the 5' end of said fragments, - a stage of addition of the reagents necessary for the enzymatic addition, a stage of enzymatic addition of Xi nucleotides to the second end of said nucleic acid fragments, the free end of the nucleotide(s) added to the fragments has a protective chemical group, to prevent multiple addition of Xi nucleotides to the same fragment, - a stage of detaching from said first support said fragments comprising n + Xi nucleotides, b) a phase of purification of the fragments having a correct sequence comprising n + Xi nucleotides, comprising the following successive stages: - a second stage of attaching, to said second support, said fragments comprising n + Xi nucleotides by their 3' end, - a stage of removal of the fragments which have not been added to and of the fragments which have not been attached to the said second support, - a stage of detaching said fragments comprising n + Xi nucleotides from said second support - a second stage of attaching, to said second support, said fragments comprising n + Xi nucleotides by their 3' end, - a stage of removal of the fragments which have not been added to and of the fragments which have not been attached to the said second support, - a stage of detaching said fragments comprising n + Xi nucleotides from said second support