  For a mobile spin-1/2 impurity, coupled antiferromagnetically to a
one-dimensional gas of fermions, perturbative ideas have been used to argue in
favor of two-channel Kondo behavior of the impurity spin. Here we combine
general considerations and extensive numerical simulations to show that the
problem displays a novel quantum phase transition between two-channel and
one-channel Kondo screening upon increasing the Kondo coupling. We construct a
ground-state phase diagram and discuss the various non-trivial crossovers as
well as possible experimental realizations.
