We performed a theoretical study on the collapse transition of a single polyelectrolyte chain induced by multivalent cations, under the framework of the mean-field theory by taking into account the effect of small mobile ions. For a sufficiently long chain, the folding transition is found to be a two-step process; from coil to segregated and from segregated to collapsed, where the segregated state means the coexistence of an elongated coil part and a collapsed compact part in a single polymer chain in the equilibrium state. These transitions are considered to reflect competition between two states with double minima in the free energy; i.e., a first-order phase transition in a mesoscopic system. With the experimental technique of single chain observation by fluorescence microscopy, we monitored the folding transition of individual giant DNAs. The results of this experiment indicate the actual existence of the two-step folding transition in single DNA molecules. The translational entropy of ions may play an important role in this process.