In this paper, we analyze the pairwise error probability (PEP) of distributed space-time codes, in which the source and the relay generate Alamouti space--time code in a distributed fashion. We restrict our attention to the space-time code construction for Protocol III in [1]. In particular, we derive two closed-form approximations for PEP when the relay is either close to the destination or source and an upper bound for any position of the relay. Using the alternative definition of $Q$-function, we can express these PEPs in terms of finite integral whose integrand is composed of trigonometric functions. We further show that with only one relay assisted source-destination link, system still achieves diversity order of two, assuming single-antenna terminals. We also perform Monte-Carlo simulations to verify the analysis.