  We experimentally and theoretically study the diffraction phase of
large-momentum transfer beam splitters in atom interferometers based on Bragg
diffraction. We null the diffraction phase and increase the sensitivity of the
interferometer by combining Bragg diffraction with Bloch oscillations. We
demonstrate agreement between experiment and theory, and a 1500-fold reduction
of the diffraction phase, limited by measurement noise. In addition to reduced
systematic effects, our interferometer has high contrast with up to 4.4 million
radians of phase difference, and a resolution in the fine structure constant of
$\delta \alpha/\alpha=0.25\,$ppb in 25 hours of integration time.
