Using an atom interferometer, we have measured the static electric polarizability of $^7$Li $\alpha =(24.33 \pm 0.16)\times10^{-30} $ m$^3$ $= 164.19\pm 1.08 $ atomic units with a $0.66$\% uncertainty. Our experiment, which is similar to an experiment done on sodium in 1995 by D. Pritchard and co-workers, consists in applying an electric field on one of the two interfering beams and measuring the resulting phase-shift. With respect to D. Pritchard's experiment, we have made several improvements which are described in detail in this paper: the capacitor design is such that the electric field can be calculated analytically; the phase sensitivity of our interferometer is substantially better, near $16$ mrad/$\sqrt{\mbox{Hz}}$; finally our interferometer is species selective it so that impurities present in our atomic beam (other alkali atoms or lithium dimers) do not perturb our measurement. The extreme sensitivity of atom interferometry is well illustrated by our experiment: our measurement amounts to measuring a slight increase $\Delta v$ of the atom velocity $v$ when it enters the electric field region and our present sensitivity is sufficient to detect a variation $\Delta v/v \approx 6 \times 10^{-13}$.