Photoluminescence of excitons bound to Al, Ga, In, and Tl acceptors in Si crystals subjected to 〈001〉, 〈111〉, or 〈110〉 uniaxial stress was studied at liquid-He temperatures with 0.0025-meV spectral resolution. The deformation-potential constants of the group-III acceptors in the ground state are (in eV) $b=\ensuremath{-}1.01\ifmmode\pm\else\textpm\fi{}0.02,$ $d=\ensuremath{-}3.31\ifmmode\pm\else\textpm\fi{}0.06$ for Al, $b=\ensuremath{-}1.03\ifmmode\pm\else\textpm\fi{}0.02,$ $d=\ensuremath{-}3.10\ifmmode\pm\else\textpm\fi{}0.06$ for Ga, $b=\ensuremath{-}0.43\ifmmode\pm\else\textpm\fi{}0.01,$ $d=\ensuremath{-}2.41\ifmmode\pm\else\textpm\fi{}0.05$ for In, and $b=\ensuremath{-}0.30\ifmmode\pm\else\textpm\fi{}0.03,$ $d=\ensuremath{-}1.95\ifmmode\pm\else\textpm\fi{}0.2$ for Tl. The shear deformation-potential constant for electrons in acceptor bound excitons ${\ensuremath{\Xi}}_{u}=8.6\mathrm{eV}$ for all group-III acceptors within an experimental error of $\ifmmode\pm\else\textpm\fi{}0.15\mathrm{eV}$ for Al, Ga, and In, and $\ifmmode\pm\else\textpm\fi{}0.8\mathrm{eV}$ for Tl. The order of the valley-orbit states in Tl bound excitons is ${\ensuremath{\Gamma}}_{1},$ ${\ensuremath{\Gamma}}_{3},$ ${\ensuremath{\Gamma}}_{5}$ with the ${\ensuremath{\Gamma}}_{5}$ energy 1.21 meV above ${\ensuremath{\Gamma}}_{1},$ and 0.10 meV above ${\ensuremath{\Gamma}}_{3}.$ All details of the spectra including positions, relative amplitudes, and polarizations of the components have been explained on the basis of a simple model of acceptor bound excitons with holes in the $J=0$ state taking into account the valley-orbit splitting and the spin-orbit coupling of the electron. Significant deviations from the theoretical predictions were observed only for very small strains producing acceptor splittings comparable with the intrinsic zero-stress splitting.