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A recent measurement of the hyperfine splitting in the ground state of Li-like 208Bi80+ has established a "hyperfine puzzle"--the experimental result exhibits a 7s 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 (µI) of 209Bi. We perform relativistic density functional theory and relativistic coupled cluster calculations of the shielding constant that should be used to extract the value of µI(209Bi) and combine it with nuclear magnetic resonance measurements of Bi(NO3)3 in nitric acid solutions and of the hexafluoridobismuthate(V) BiF-6 ion in acetonitrile. The result clearly reveals that µI(209Bi) 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. [ABSTRACT FROM AUTHOR]