FullSubNet is our recently proposed real-time single-channel speech enhancement network that achieves outstanding performance on the Deep Noise Suppression (DNS) Challenge dataset. A number of variants of FullSubNet have been proposed recently, but they all focus on the structure design towards better performance and are rarely concerned with computational efficiency. This work proposes a new architecture named Fast FullSubNet dedicated to accelerating the computation of FullSubNet. Specifically, Fast FullSubNet processes sub-band speech spectra in the mel-frequency domain by using cascaded linear-to-mel full-band, sub-band, and mel-to-linear full-band models such that frequencies involved in the sub-band computation are vastly reduced. After that, a down-sampling operation is proposed for the sub-band input sequence to further reduce the computational complexity along the time axis. Experimental results show that, compared to FullSubNet, Fast FullSubNet has only 13% computational complexity and 16% processing time, and achieves comparable or even better performance.