Though there are intensive researches on off-line electronic cash (e-cash), the current computer network infrastructure sufficiently accepts on-line e-cash. The on-line means that the payment protocol involves with the bank, and the off-line means no involvement. For customers' privacy, the e-cash system should satisfy unlinkability, i.e., any pair of payments is unlinkable w.r.t. the sameness of the payer. In addition, for the convenience, exact payments, i.e., the payments with arbitrary amounts, should be also able to performed. In an existing offline system with unlinkable exact payments, the customers need massive computations. On the other hand, an existing on-line system does not satisfy the efficiency and the perfect unlinkability simultaneously. This paper proposes an on-line system, where the efficiency and the perfect unlinkability are achieved simultaneously.