A simple CFAR detector is proposed in this paper for detecting complex sinusoidal signals with unknown parameters in complex Gaussian noise with unknown variance. The detector is based on the locally most powerful test (LMP), which performance approaches the uniformly most powerful test (UMP) when the signal-to-noise ratio (SNR) approaches zero. It can maximize the probability of detection for a given false alarm rate, especially when the signal is weak. We adopt the estimate and plug detector method to complete the design. The performance analysis shows that the detector we designed is an approximate constant false alarm rate (CFAR) detector. Compared with the generalized likelihood ratio test (GLRT), it is easier to complete. Simulation results support our conclusion.