  We consider the possibility of the coupling constants of the $SU(3)\times
SU(2)\times U(1)$ gauge interactions at the time of big bang nucleosynthesis
having taken different values from what we measure at present, and investigate
the allowed difference requiring the shift in the coupling constants not
violate the successful calculation of the primordial abundances of the light
elements. We vary gauge couplings and Yukawa couplings (fermion masses) using a
model in which their relative variations are governed by a single scalar field,
dilaton, as found in string theory. The results include a limit on the fine
structure constant
$-6.0\times10^{-4}<\Delta\alpha_{EM}/\alpha_{EM}<1.5\times10^{-4}$, which is
two orders stricter than the limit obtained by considering the variation of
$\alpha_{EM}$ alone.
