  The logarithmically enhanced alpha^7 ln(1/alpha) contribution to the
hyperfine splitting of the positronium ground-state energy levels is calculated
in the framework of dimensionally regularized nonrelativistic quantum
electrodynamics. The correction is negative and amounts to about 1/3 of the
leading logarithmic alpha^7 ln^2(1/alpha) one. The discrepancy between the
experimental measurements and the theoretical prediction is reduced.
