  A systematic perturbation theory is developed for the ESR shift and is
applied to the spin-1/2 Heisenberg chain. Using the Bethe ansatz technique, we
exactly analyze the resonance shift in the first order of perturbative
expansion with respect to an anisotropic exchange interaction. Exact result for
the whole range of temperature and magnetic field, as well as asymptotic
behavior in the low-temperature limit are presented. The obtained g-shift
strongly depends on magnetic fields at low temperature, showing a significant
deviation from the previous classical result.
