With the rapid development of cloud storage, more and more cloud clients can store and access their data anytime, from anywhere and using any device. Data deduplication may be considered an excellent choice to ensure data storage efficiency. Although cloud technology offers many advantages for storage service, it also introduces security challenges, especially with regards to data integrity, which is one of the most critical elements in any system. A data owner should thus enable data integrity auditing mechanisms. Much research has recently been undertaken to deal with these issues. In this paper, we propose a novel blockchain-based method, which can preserve cloud data integrity checking with data deduplication. In our method, a mediator performs data deduplication on the client side, which permits a reduction in the amount of outsourced data and a decrease in the computation time and the bandwidth used between the enterprise and the cloud service provider. This method supports private and public auditability. Our method also ensures the confidentiality of a client's data against auditors during the auditing process.