The detection of weak signals is a well-established application in chaos theory. This theory leverages the inherent robustness of chaotic systems, enabling them to resist noise and thus serve as effective tools for identifying weak signals. However, challenges remain in selecting appropriate chaotic systems and in their practical implementation—areas that are still under-explored. In this paper, we analyze a simplified fractional-order Genesio–Tesi chaotic system, which exhibits a unique chaos-divergence characteristic. Based on this characteristic, we propose a new detection method that uses the chaos-divergence state as a criterion for determining the presence or absence of a signal when detecting weak signal amplitudes. This approach makes the simplified fractional-order Genesio–Tesi chaotic system more suitable for chaotic weak signal detection. Notably, the significant variance observed in the divergent state’s independent variables emerges as a key feature, enhancing the system’s ability to detect the frequencies of weak signals. Our numerical simulations focus on detecting weak cosine signals masked by three different types of noise. The results demonstrate successful detection of a weak signal at a frequency of 100 rad/s under the specified conditions, with the lowest detectable signal-to-noise ratio of −40.83 dB. Overall, these results highlight the effectiveness and feasibility of our proposed method for weak signal detection.