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We present a method for the detection of action potentials, an essential first step in the analysis of extracellular neural signals. The low signal-to-noise ratio (SNR) and similarity of spectral characteristic between the target signal and background noise are obstacles to solving this problem and, thus, in previous studies on experimental neurophysiology, only action potentials with sufficiently large amplitude have been detected and analyzed. In order to lower the level of SNR required for successful detection, we propose an action potential detector based on a prudent combination of wavelet coefficients of multiple scales and demonstrate its performance for neural signal recording with varying degrees of similarity between signal and noise. The experimental data include recordings from the rat somatosensory cortex, the giant medial nerve of crayfish, and the cutaneous nerve of bullfrog. The proposed method was tested for various SNR values and degrees of spectral similarity. The method was superior to the Teager energy operator and even comparable to or better than the optimal linear detector. A detection ratio higher than 80% at a false alarm ratio lower than 10% was achieved, under an SNR of 2.35 for the rat cortex data where the spectral similarity was very high.