In this paper, we consider an uncoded cooperative relaying system using an adaptive decode-and-forward (ADF) protocol. Unlike many existing works that adopt single-symbol transmissions, we assume more reasonable frame transmissions here. The primary focus of this paper is on the performance aspects, aiming to offer important theoretical guidance to system designers. Specifically, we derive an accurate bit error rate (BER) expression as well as its asymptotic approximation for the considered ADF cooperative system. The presented analysis shows that the diversity gain can be fully achieved for arbitrary frame length, while the coding gain depends on the frame length. Moreover, by maximizing the coding gain, we obtain an optimum power allocation (OPA) scheme, which can provide a power gain of up to 3 dB over the equal power allocation (EPA) scheme. Simulation results are presented to verify the theoretical analysis.