We extract the value of the strong coupling constant \alpha _s from a single-parameter pointlike fit to the photon structure function {F₂^\gamma } at large x and Q² and from a first five-parameter full (pointlike and hadronic) fit to the complete {F₂^\gamma } data set taken at PETRA, TRISTAN, and LEP. In next-to-leading order and the {\overline {MS}} renormalization and factorization schemes, we obtain \alpha _s(m_Z)=0.1183\pm 0.0050(exp.)^+0.0029_-0.0028(theor.) [pointlike] and \alpha _s(m_Z)=0.1198\pm 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 {F₂^\gamma } is now possible.

PACS numbers: 12.38.Bx, 12.38.Qk, 13.65.+i