Minimum mean-square error (MMSE) approaches to speech enhancement are widely used in the literature. The quality of enhanced speech produced by an MMSE approach is directly impacted by the accuracy of the employed a priori signal-to-noise ratio (SNR) estimator. In this paper, the a priori SNR estimate spectral distortion (SD) level that results in a just-noticeable difference (JND) in the perceived quality of MMSE approach enhanced speech is found. The JND SD level is indicative of the accuracy that an a priori SNR estimator must exceed to have no impact on the perceived quality of MMSE approach enhanced speech. To measure the JND SD level, listening tests are conducted across five SNR levels, five noise sources, and two MMSE approaches [the MMSE short-time spectral amplitude (MMSE-STSA) estimator and the Wiener filter]. A statistical analysis of the results indicates that the JND SD level increases with the SNR level, is higher for the MMSE-STSA estimator, and is not impacted by the type of background noise. Following the literature, a significant improvement in a priori SNR estimation accuracy is required to reach the JND SD level.