Order alpha(3) ln(1/alpha) corrections to positronium decays

The logarithmically enhanced alpha(3)ln(1/alpha) corrections to the para- and orthopositronium decay widths are calculated in the framework of dimensionally regularized nonrelativistic quantum electrodynamics. In the case of parapositronium, the correction is negative, approximately doubles the effect of the leading logarithmic alpha(3)ln (2)(1/alpha) one, and is comparable to the nonlogarithmic O(alpha(2)) one. As for orthopositronium, the correction is positive and almost cancels the alpha(3)ln (2)(1/alpha) one. The uncertainty in the theoretical prediction for the parapositronium decay width is reduced to 10(-2) &mgr;s(-1).