Relay-assisted cooperative communication has received considerable attention as a promising approach to alleviate the burden of installing multiple antennas on simple terminals but still achieving transmit diversity. In most prior work, perfect error detection is assumed to prevent relays that receive in error from forwarding. Since error detection is often performed at the medium access control (MAC) layer, each relay in the network needs to pass the received signal from the physical layer to the MAC layer before making the forwarding decision, resulting in the overhead that may not be necessary. Selection relaying thus has been proposed to avoid the MAC overhead at relays. The required election phase in selection relaying, however, needs to last for a deterministic duration proportional to the number of relays in a coordinated manner, or a nondeterministic duration in the absence of coordination. In this paper, we propose a cooperative relaying scheme that achieves diversity gain with a constant delay that is insensitive to the network size. In the proposed scheme, relays perform threshold testing instead of error detection, and a fix duration is allocated for the election phase. We analyze the end-to-end bit error rate (BER) of the proposed scheme in closed-form expressions, and prove the achievable diversity order. Numerical results are given to demonstrate the superior performance of the proposed scheme over the direct retransmission and the well-known opportunistic relaying scheme in terms of the end-to-end BER, average delay and delay variance.