  We extract the value of the strong coupling constant alpha_s from a
single-parameter pointlike fit to the photon structure function F_2^gamma at
large x and Q^2 and from a first five-parameter full (pointlike and hadronic)
fit to the complete F_2^gamma data set taken at PETRA, TRISTAN, and LEP. In
next-to-leading order and the MSbar renormalization and factorization schemes,
we obtain alpha_s(m_Z)=0.1183 +/- 0.0050(exp.)^+0.0029_-0.0028(theor.)
[pointlike] and alpha_s(m_Z)=0.1198 +/- 0.0028(exp.)^+0.0034_-0.0046(theor.)
[pointlike and hadronic]. We demonstrate that the data taken at LEP have
reduced the experimental error by about a factor of two, so that a competitive
determination of alpha_s from F_2^gamma is now possible.
