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 $10$ mrad/$\sqrt{\mbox{Hz}}$; finally our interferometer is species selective: impurities present in our atomic beam (other 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 $\Delta v/v \approx 6 \times 10^{-13}$.