A possibility for a determination of the fine structure constant in experiments on the bound-electron g-factor is examined. It is found that studying a specific difference of the g-factors of B- and H-like ions of the same spinless isotope in the Pb region to the currently accessible experimental accuracy of 7 x 10(-10) would lead to a determination of the fine structure constant to an accuracy which is better than that of the currently accepted value. Further improvements of the experimental and theoretical accuracy could provide a value of the fine structure constant which is several times more precise than the currently accepted one.