We consider blind late-reverberation suppression in speech signals measured with a single microphone in noisy environments. We exploit that reverberant speech shows correlation over longer time spans than clean speech by predicting the contribution of reverberant energy to the current observed spectrum from the enhanced spectra of previous frames. The prediction parameters are recursively updated with estimates of the correlation coefficients between the current reverberant spectrum and enhanced previous spectra. The contributions of late reverberation and noise are suppressed by a standard noise reduction algorithm. The algorithm is shown to decrease the long-term correlation. It achieves significant improvements in segmental speech-to-interference ratio and Bark spectral distortion for typical reverberation times and noise levels, while almost no distortions are introduced in clean speech.