  We put bounds on the variation of the value of the fine structure constant
$\alpha$, at the time of Big Bang nucleosynthesis. We study carefully all light
elements up to $^7$Li. We correct a previous upper limit on $|\Delta \alpha /
\alpha|$ estimated from $^4$He primordial abundance and we find interesting new
potential limits (depending on the value of the baryon-to-photon ratio) from
$^7$Li, whose production is governed to a large extent by Coulomb barriers. The
presently unclear observational situation concerning the primordial abundances
preclude a better limit than $|\Delta \alpha/\alpha| \lsim 2\cdot 10^{-2}$, two
orders of magnitude less restrictive than previous bounds. In fact, each of the
(mutually exclusive) scenarios of standard Big Bang nucleosynthesis proposed,
one based on a high value of the measured deuterium primordial abundance and
one based on a low value, may describe some aspects of data better if a change
in $\alpha$ of this magnitude is assumed.
