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It is well known that the "musical noise" encountered in most frequency domain speech enhancement algorithms is partially due to the large variance estimates of the spectra. To address this issue, we propose in this paper the use of low-variance spectral estimators based on wavelet thresholding the multitaper spectra for speech enhancement. A short-time spectral amplitude estimator is derived which incorporates the wavelet-thresholded multitaper spectra. Listening tests showed that the use of multitaper spectrum estimation combined with wavelet thresholding suppressed the musical noise and yielded better quality than the subspace and MMSE algorithms.