  We present an analytic calculation of the m*alpha^6 recoil corrections to the
hyperfine splitting (HFS) of the ground state energy levels in positronium. We
find \Delta E_{\rm rec} = m alpha^6 (-1/6 ln(alpha) + 331/432 -ln(2)/4 - 17
Zeta(3)/(8 pi^2) + 5/(12 pi^2)) \approx m alpha^6 (- 1/6 ln(alpha) + 0.37632),
confirming Pachucki's numerical result \cite{Ph}. We present a complete
analytic formula for the m*alpha^6 HFS of the positronium ground state and,
including m*alpha^7*ln^2(alpha) effects, find E(1^3S_1)-E(1^1S_0))_{theory} =
203 392(1) MHz. This differs from the experimental results by about 3 standard
deviations.
