A method for amplifying desired nucleic acid molecules by PCR which comprises the steps of isolating and purifying a group of nucleic acid molecules including desired nucleic acid molecules to be amplified, then carrying out PCR while establishing such a condition that the nucleic acid molecules capable of being amplified, except for primers, present in a PCR reaction solution are constituted by only the desired nucleic acid molecules and that the concentration of each primer used is limited to a level of not more than 100 nM; a method for producing a protein in a cell-free protein-synthesis system containing a cell-free extract which comprises using the nucleic acid of a single kind produced by the method described above and a method for establishing a protein library which comprises the steps of separately carrying out the method for amplifying desired nucleic acid molecules by PCR to obtain at least two kinds of nucleic acid molecules and separately carrying out the foregoing method for producing a protein using each of the at least two kinds of the amplified nucleic acids as a template to thus establish a protein library which comprises at least two kinds of proteins encoded by the resulting at least two nucleic acids respectively.