  A recent measurement of the hyperfine splitting in the ground state of
Li-like $^{209}$Bi$^{80+}$ has established a "hyperfine puzzle" -- the
experimental result exhibits a 7$\sigma$ deviation from the theoretical
prediction [J. Ullmann et al., Nat. Commun. 8, 15484 (2017); J. P. Karr, Nat.
Phys. 13, 533 (2017)]. We provide evidence that the discrepancy is caused by an
inaccurate value of the tabulated nuclear magnetic moment ($\mu_I$) of
$^{209}$Bi. We perform relativistic density functional theory and relativistic
coupled cluster calculations of the shielding constant that should be used to
extract the value of $\mu_I(^{209}{\rm Bi})$ and combine it with nuclear
magnetic resonance measurements of Bi(NO$_3$)$_3$ in nitric acid solutions and
of the hexafluoridobismuthate(V) BiF$_6^-$ ion in acetonitrile. The result
clearly reveals that $\mu_I(^{209}{\rm Bi})$ is much smaller than the tabulated
value used previously. Applying the new magnetic moment shifts the theoretical
prediction into agreement with experiment and resolves the hyperfine puzzle.
