The hadronic photon structure function Fγ2 is studied in the reaction e+e− → e+e−hadrons at LEP with the L3 detector. The data, collected from 1991 to 1995 at a centre-of-mass energy √s = ≃ 91 GeV, correspond to an integrated luminosity of 140 pb−1. The photon structure function Fγ2 is measured in the Q2 interval 1.2 GeV2 ≤ Q2 ≤ 9.0 GeV2 and the x interval 0.002 < x < 0.2. Fγ2 shows a linear growth with lnQ2. The value of the slope α−1dFγ2(Q2)/dlnQ2 is measured to be 0.079±0.011±0.009.