Most current wireless IEEE 802.11 networks rely on a power-threshold based carrier-sensing multi-access (CSMA) mechanism to prevent packet collisions, in which a transmitter permits its transmission only if the locally measured aggregate interference power from all existing transmissions is below a prespecified power-sensing threshold. However, such a mechanism can not completely guarantee interference-safe transmissions, leading to the so-called hidden-node problem, which causes degradation in throughput and fairness performance. Traditionally, ensuring interference-safe transmissions was addressed by simple models of conflict graphs, rather than by the realistic signal-to-interference-and-noise ratio (SINR) model. This paper presents the first viable solution for fully interference-safe transmissions that (1) assumes an accurate SINR model, and (2) is compatible with the carrier-sensing mechanism in existing CSMA networks. Specifically, we determine a proper interference-safe power-sensing threshold by considering both the effects of (i) arbitrary ordering of local interference power measurements, and (ii) ACK frames. We compare our interference-safe solution with other solutions, and provide extensive evaluation on its throughput and fairness performance.