  We construct an $O(a^2)$-improved overlap-Dirac operator by designing an
improved overlap kernel, based on the Symanzik improvement program. Field
rotation terms are also identified to improve off-shell amplitudes for both
massless and massive fermions. We check the free dispersion relation and
propagator, and show that improved results become to close to the continuum
ones at low momentum region. We test the effect of improvement on the full-QCD
gauge configuration and find that the relativistic dispersion relation is
satisfied within a few percent error up to $m_qa \approx 0.5$.
