The hyperfine structure splitting in the 6p34S3/2→6p27s 4P1/2transition at 307nm in atomic 208Bi was measured with collinear laser spectroscopy at ISOLDE, CERN. The hyperfine Aand Bfactors of both states were determined with an order of magnitude improved accuracy. Based on these measurements, theoretical input for the hyperfine structure anomaly, and results from hyperfine measurements on hydrogen-like and lithium-like 209Bi80+,82+, the nuclear magnetic moment of 208Bi has been determined to μ(208Bi) =+4.570(10) μN. Using this value, the transition energy of the ground-state hyperfine splitting in hydrogen-like and lithium-like 208Bi80+,82+and their specific difference of −67.491(5)(148)meV are predicted. This provides a means for an experimental confirmation of the cancellation of nuclear structure effects in the specific difference in order to exclude such contributions as the cause of the hyperfine puzzle, the recently reported 7-σdiscrepancy between experiment and bound-state strong-field QED calculations of the specific difference in the hyperfine structure splitting of 209Bi80+,82+.