We propose a novel phase-based method for single-channel speech enhancement to extract and enhance the desired signals in noisy environments by utilizing the phase information. In the method, a phase-dependent a priori signal-to-noise ratio (SNR) is estimated in the log-mel spectral domain to utilize both the magnitude and phase information of input speech signals. The phase-dependent estimator is incorporated into the conventional magnitude-based decision-directed approach that recursively computes the a priori SNR from noisy speech. Additionally, we reduce the performance degradation owing to the one-frame delay of the estimated phase-dependent a priori SNR by using a minimum mean square error (MMSE)-based and maximum a posteriori (MAP)-based estimator. In our speech enhancement experiments, the proposed phase-dependent a priori SNR estimator is shown to improve the output SNR by 2.6 dB for both the MMSE-based and MAP-based estimator cases as compared to a conventional magnitude-based estimator.