  The problem of combating de-coherence by weak measurements has already been
studied for the amplitude damping channel and for specific input states. We
generalize this to a large four-parameter family of qubit channels and for the
average fidelity over all pure states. As a by-product we classify all the
qubit channels which have one invariant pure state and show that the parameter
manifold of these channels is isomorphic to $S^2\times S^1\times S^1$ and
contains many interesting subclasses of channels. The figure of merit that we
use is the average input-output fidelity which we show can be increased up to
$30$ percents in some cases, by tuning of the weak measurement parameter.
