A fast-decodable distributed space-time block code (STBC) is proposed for a two-path successive relay network that can achieve both full diversity and full transmission rate. The proposed STBC employs a precoder to rotate the constellation of transmitted symbols at source. After decode-and-forward by relays, the rotated information symbols are decoded at destination. It is shown that single complex symbol decoding can be obtained at the destination. Simulation results demonstrate that the full diversity is offered by the newly proposed distributed STBC for the two-path successive relay network.