We have measured the photon structure function F2gamma in the reaction e+e- --> e+e- hadrons for average Q2 values from 5.1 to 338 GeV2 by using data collected by the TOPAZ detector at TRISTAN. The data have been corrected for detector effects and are compared with theoretical expectations based on QCD. The structure function F2gamma increases as In Q2, as expected. A sample of events with one or two distinct jets has been identified in the final state. Although two-jet events can be explained solely by the point-like perturbative part, one-jet events require a significant hadron-like part in addition.