  We obtain an improved determination of the normalization of the leading
infrared renormalon of the Bjorken sum rule: $Z_B (n_f=3)= -0.407\pm 0.119 $.
Estimates of higher order terms of the perturbative series are given. We
compute the Bjorken sum rule with hyperasymptotic precision by including the
leading terminant, associated with the first infrared renormalon. We fit the
experimental data to the operator product expansion theoretical prediction with
$\hat f_{3}^{\rm PV}$ as the free parameter. We obtain a good agreement with
the experiment with $\hat f_{3}^{\rm PV}\times 10^3=32^{+187}_{-196}\;{\rm
GeV}^{2}$ for $Q^2 \geq 1$ GeV$^2$.
