  We present a new estimate of the fine structure constant and the
$\beta$-function of QED at an arbitrary scale. Using the non-perturbative but
convergent series expression of the one loop effective action of QED that has
been available recently we make a non-perturbative estimate of the running
coupling and the $\beta$-function, and prove the renormalization group
invariance of the effective action. The contrast between our result and the
perturbative result is remarkable.
