  We study the evolution of a qubit evolving according to the Schr\"odinger
equation with a Hamiltonian containing noise terms, modeled by random diagonal
and off-diagonal matrix elements. We show that the noise-averaged qubit density
matrix converges to a final state, in the limit of large times $t.$ The
convergence speed is polynomial in $1/t$, with a power depending on the
regularity of the noise probability density and its low frequency behaviour. We
evaluate the final state explicitly. We show that in the regimes of weak and
strong off-diagonal noise, the process implements the dephasing channel in the
energy- (localized) and the delocalized basis, respectively.
