  Big Bang Nucleosynthesis (BBN) is very sensitive to the cosmological
expansion rate. If the gravitational constant $G$ took a different value during
the nucleosynthesis epoch than today, the primordial abundances of light
elements would be affected. In this work, we improve the bounds on this
variation using recent determinations of the primordial element abundances,
updated nuclear and weak reaction rates and observations of the Cosmic
Microwave Background (CMB). When combining the measured abundances and the
baryon density from CMB observations by Planck, we find $G_\mathrm{BBN}/G_0 =
0.99^{+0.06}_{-0.05}$ at $2\sigma$ confidence level. If the variation of $G$ is
linear in time, we find $\dot{G}/G_0 = 0.7^{+3.8}_{-4.3}\times 10^{-12} \,
\mathrm{yr}^{-1}$, again at $2\sigma$. These bounds are significantly stronger
than those from previous primordial nucleosynthesis studies, and are comparable
and complementary to CMB, stellar, solar system, lunar laser ranging, pulsar
timing and gravitational wave constraints.
