The process for varying the speed of a speech signal that involves splitting at least a portion of the speech frequency bandwidth into N narrow sub-bands, processing each sub-hand signal contents to derive therefrom magnitude data M(i, n) and phase data P(i, n), i=1, . . . , N being the subband index and n the time index. The M (i, n) sequence is converted into a sequence M'(n) by either duplicating one sample every K samples (K being an integer value derived from the desired slowing-down/speeding up ratio). The phase sequence P (i, n) is processed to derive therefrom an increment sequence D(i, n)=P(i, n)-P(i, n-1), which increment sequence is first converted into a D'(i, n) sequence by either dropping or duplicating one sample every K, samples, before being converted into P'(i, n)=P'(i, n)+D'(i, n). The P'(i, n), D'(i, n) sequences are converted back into sub-band signals contents, then combined together into the slowed-down/speeded-up speech signal.