  In a conformal invariant one-dimensional stochastic model, a certain
non-local perturbation takes the system to a new massless phase of a special
kind. The ground-state of the system is an adsorptive state. Part of the
finite-size scaling spectrum of the evolution Hamiltonian stays unchanged but
some levels go exponentially to zero for large lattice sizes becoming
degenerate with the ground-state. As a consequence one observes the appearance
of quasistationary states which have a relaxation time which grows
exponentially with the size of the system. Several initial conditions have
singled out a quasistationary state which has in the finite-size scaling limit
the same properties as the stationary state of the conformal invariant model.
