  We study the low-temperature properties of a spin-\onehalf\ magnetic impurity
coupled to a one-dimensional interacting electron system. Using the newly
developed formalism by Affleck and Ludwig, with a scale invariant boundary
condition replacing the impurity, we exploit boundary conformal field theory to
deduce the impurity thermal and magnetic response. In the case of only forward
electron scattering off the impurity, we predict the same critical scaling as
for the two-channel Kondo effect for non-interacting electrons, but with a
novel Wilson ratio. Backward electron scattering off the impurity destabilizes
this behavior and drives the system to a new fixed point. In the case of equal
amplitudes for forward- and backward scattering {\em (Kondo interaction)}, we
show that there are only two types of scaling behaviors consistent with the
symmetries of the problem: {\em either} a local Fermi liquid {\em or} a
critical theory with an anomalous specific heat. The latter case agrees with a
recent ``poor-man-scaling'' result proposed by Furusaki and Nagaosa.
