The invention is directed to a method for obtaining a nucleic acid library of a sample comprising polynucleotides comprising the steps a. multiplying the polynucleotides by a polymerase b. fragmenting the multiplied polynucleotides by creating nicks c. coupling an oligonucleotide sequence to the nicks to create the target library wherein step a) is performed by providing A, T, G, C and U nucleotides wherein the molar ratio of T and U is between 200:1 and 5:1 and step b) is performed by excision of the U nucleotides. characterized in that after step b), the nicks are provided with a polymerase exhibiting 5′ #3′ exonuclease activity, thereby filling in the 3′ recessing ends and removing the 5′ overhangs of the nicks.