The hyperfine structure splitting in the 6p ³S3/24→6p ²7sP1/24 transition at 307 nm in atomic ²⁰⁸Bi was measured with collinear laser spectroscopy at ISOLDE, CERN. The hyperfine A and B factors 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 ²⁰⁹Bi ^80+,82+, the nuclear magnetic moment of ²⁰⁸Bi has been determined to μ(Bi208)=+4.570(10)μ _N. Using this value, the transition energy of the ground-state hyperfine splitting in hydrogen-like and lithium-like ²⁰⁸Bi ^80+,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-σ discrepancy between experiment and bound-state strong-field QED calculations of the specific difference in the hyperfine structure splitting of ²⁰⁹Bi ^80+,82+.