The Zeeman effect on bound excitons in Si doped with boron has been studied in magnetic fields of up to 12 T, using Fourier-transform photoluminescence spectroscopy with a resolution of 3 \ensuremath{\mu}eV. Up to 20 narrow spectral components of the no-phonon boron-bound-exciton line have been resolved in each of the three 〈001〉, 〈111〉, and 〈110〉 sample orientations. In addition to the linear paramagnetic splitting of spectral components, a quadratic diamagnetic splitting was observed, and was attributed to the difference in the diamagnetic shifts of the single-electron states associated with the different conduction-band minima. From the pattern of the bound-exciton splittings, the order of the valley-orbit energy levels has been determined to be ${\mathrm{\ensuremath{\Gamma}}}_{3}$,${\mathrm{\ensuremath{\Gamma}}}_{5}$,${\mathrm{\ensuremath{\Gamma}}}_{1}$, with level ${\mathrm{\ensuremath{\Gamma}}}_{3}$ being the lowest and ${\mathrm{\ensuremath{\Gamma}}}_{1}$ the highest. A perturbation Hamiltonian, constructed from symmetry considerations, and describing the valley-orbit splitting, interparticle correlations, and interactions with the magnetic field, was used for calculations of the boron-bound-exciton energy levels versus field. Phenomenological parameters, including interparticle-correlation constants, g factors, and diamagnetic-shift constants were determined by simultaneously optimizing the fit between experimentally observed and calculated energy levels in strong magnetic fields and under uniaxial stress.