This paper proposes a new speech enhancement (SE) algorithm utilizing constraints to the Wiener gain function which is capable of working at 10 dB and lower signal-to-noise ratios (SNRs). The wavelet thresholded multitaper spectrum was taken as the clean spectrum for the constraints. The proposed algorithm was evaluated under eight types of noises and seven SNR levels in NOIZEUS database and was predicted by the composite measures and the SNRLOSS measure to improve subjective quality and speech intelligibility in various noisy environments. Comparisons with two other algorithms (KLT and wavelet thresholding (WT)) demonstrate that in terms of signal distortion, overall quality, and the SNRLOSS measure, our proposed constrained SE algorithm outperforms the KLT and WT schemes for most conditions considered.