The quantum key distribution (QKD) network [1,2] with Vernam's One Time Pad (OTP) encryption [3] and secret sharing [4] are powerful security tools to realize an information theoretically secure (ITS) distributed storage system. In [5], a single-password-authenticated secret sharing (SPSS) scheme based on the QKD network and Shamir's secret sharing [4] was experimentally demonstrated; it confirmed ITS data transmission, storage, authentication, and integrity. To achieve data integrity, an ITS message authentication code (MAC) tag is employed and a data owner of the secret sharing performs both the MAC tag generation and verification. However, for a scenario in which the data owner and end users are different entities, the above approach may not work since the data owner can cheat the end users. In this paper, we resolve this problem by proposing an ITS integrity protection scheme employing a third-party verification with timestamp. The ITS integrity protection is realized by two steps: integrity check by the data owner at data reconstruction, and data integrity certification by the data owner, the end user, and the third-party verifier using a MAC based on universal2 hash function [6] and random number provided from the QKD network [7]. In addition to introduce the third-party verifier, we institute "a trusted calculator" which computes shares of the data and MAC tags and sends MAC tags to the third-party verifier. The random number used in calculating MAC tag is stored in the trusted calculator. We implement this scheme on the SPSS system installed in the Tokyo QKD Network [8] and evaluate the performance of the third-party verification in view of attack scenarios on this system. In addition, we demonstrate a simple share renewal function based on verifiable secret sharing scheme which ensures the data integrity for a certain practical period based on the hardness of discrete logarithm problem. To our best knowledge, this is the first demonstration of ITS secure data transmission, storage, authentication, and data integrity with the third-party in a metropolitan area network.