  The movement of a purely elastic interface driven on a disordered energy
potential is characterized by a depinning transition: when the pulling force S
is larger than some critical value S_1 the system is in a flowing regime and
moves at a finite velocity. If S < S_1 the interface remains pinned and its
velocity is zero. We show that for a one-dimensional interface, the inclusion
of viscoelastic relaxation produces the appearance of an intervening regime
between the pinned and the flowing phases in a well defined stress interval
S_0<S<S_1, in which the interface evolves through a sequence of avalanches that
give rise to a creep process. As S --> S_0 the creep velocity vanishes in an
universal way that is governed by a directed percolation process. As S --> S_1
the creep velocity increases as a power law due to the increase of the typical
size of the avalanches. The present observations may serve to improve the
understanding of fatigue failure mechanisms.
