We present a new method to study the melting transition of DNA oligonucleotides, which can quantify the presence of intermediate states. The approach is to combine UV spectroscopy with a method based on trapping intermediate states by quenching. The measurements yield both the average fraction of open base pairs (f) and the fraction of completely open molecules (p). If intermediate (partially open) states are not present, then p = f throughout the transition. In the presence of intermediate states, p < f. We demonstrate the method on the example of a 48mer sequence which is designed to open at one end and thus have intermediate states during melting. Then we show a different sequence design where the melting appears essentially as a two-states process. These experiments demonstrate the role played by end effects and sequence design in controlling the nature of the melting transition for DNA oligomers.