Sequencing of a selected region of a target nucleic acid polymer in a genomic DNA sample can be performed in a single vessel by combining the sample with a sequencing mixture containing a primer pair, a thermally stable polymerase such as Thermo Sequenase.TM. which incorporates dideoxynucleotides into an extending nucleic acid polymer at a rate which is no less than about 0.4 times the rate of incorporation of deoxynucleotides, nucleotide feedstocks, and a chain terminating nucleotide. The mixture is processed through multiple thermal cycles for annealing, extension and denaturation to produce a product mixture which is analyzed by electrophoresis.