  We use the Density Matrix Renormalization Group method to study a
one-dimensional chain with Peierls electron-phonon coupling, which describes
the modulation of the electron hopping by lattice distortions. We demonstrate
that this system is stable against phase separation in the dilute density
limit. We only find phase separation numerically for large couplings for which
the linear approximation for the electron-phonon coupling becomes invalid; this
behavior can be stabilized in a narrow sliver of the physical parameter space
if the dispersion of the phonons is carefully tuned. These results indicate
that in the dilute electron density limit, Peierls bipolaron liquids are
generically stable, unlike in other models of electron-phonon coupling. We show
that this behavior extends to finite carrier concentrations of up to quarter
filling. This stability of low-density, light-mass bipolaron liquids in the
Peierls model opens a path to high-$T_c$ superconductivity based on a
bipolaronic mechanism, in higher dimensions.
