  We establish existence with sharp rates of decay and distance from the
Chapman--Enskog approximation of small-amplitude quasilinear relaxation shocks
in the general case that the profile ODE may become degenerate. Our method of
analysis follows the general approach used by M\'etivier and Zumbrun in the
semilinear case, based on Chapman--Enskog expansion and the macro--micro
decomposition of Liu and Yu. In the quasilinear case, however, we find it
necessary to apply a parameter-dependent Nash-Moser iteration to close the
analysis, whereas, in the semilinear case, a simple contraction-mapping
argument sufficed.
