  The fine structure constant $\alpha$ has a particular status in physics. Its
precise determination is required to test the quantum electrodynamics (QED)
theory. The constant $\alpha$ is also a keystone for the determination of other
fundamental physical constants, especially the ones involved in the framework
of the future International System of units. This paper presents Paris
experiment, where the fine structure constant is determined by measuring the
recoil velocity of a rubidium atom when it absorbs a photon. The impact of the
recent improvement of QED calculations of the electron moment anomaly and the
recent measurement of the cesium atom recoil at Berkeley will be discussed. The
opportunity to provide a precise value of the ratio $h/m_{\mathrm{u}}$ between
the Planck constant and the atomic mass constant will be investigated.
