  We report on exact results for the low-temperature thermodynamics of a
spin-$\frac{1}{2}$ magnetic impurity coupled to a one-dimensional interacting
electron system. By using boundary conformal field theory, we show that there
are only two types of critical behaviors consistent with the symmetries of the
problem: {\em either} a local Fermi liquid, {\em or} a theory with an anomalous
response identical to that recently proposed by Furusaki and Nagaosa.
Suppression of back scattering off the impurity leads to the same critical
properties as for the two-channel Kondo effect.
