We present the details of a calculation of the mα6lnα-1 contributions to the triplet-singlet splitting of the positronium ground state. Our result Δν(α6lnα-1)=34α4Rlnα-1=34 MHz places the theoretical value for the transition frequency ν approximately 1 standard deviation above the most recent experimental value. The calculation is performed using a perturbation theory based on the Bethe-Salpeter equation and on a wave function obtained by a single iteration from the nonrelativistic Pauli wave function. After a study of all relevant terms, we conclude that the only diagrams which contribute to our order are those involving the exchange of one or two virtual transverse photons. These contributions are explicitly evaluated, and we show that other possible contributions actually vanish.