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)/(8pi^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.
