  In the absence of a half-bound state, a compactly supported potential of a
Schr\"odinger operator on the line is determined up to a translation by the
zeros and poles of the meropmorphically continued left (or right) reflection
coefficient. The poles are the eigenvalues and resonances, while the zeros also
are physically relevant. We prove that all compactly supported potentials
(without half-bound states) that have reflection coefficients whose zeros and
poles are $\eps$-close in some disk centered at the origin are also close (in a
suitable sense). In addition, we prove stability of small perturbations of the
zero potential (which has a half-bound state) from only the eigenvalues and
resonances of the perturbation.
