We consider the motion of a spin-1/2 impurity in a one-dimensional gas of spin-1/2 fermions. For antiferromagnetic interaction between the impurity and the fermions, the low temperature behavior of the system is governed by the two-channel Kondo effect, leading to the impurity becoming completely opaque to the spin excitations of the gas. As well as the known spectral signatures of the two-channel Kondo effect, we find that the low temperature mobility of the resulting “Kondo polaron” takes the universal form \(\mu\to 3\hbar v_F^2/2\pi k_B^2T^2\) in sharp contrast to the spinless case where \(\mu\propto T^{-4}\).