Double-tag events in two-photon collisions are studied using the L3 detector at LEP centre-of-mass energies from $\sqrt{s}=189~\mathrm{GeV}$ to 209~GeV. The cross sections of the $\mathrm{e^+e^-} \rightarrow \mathrm{e^+e^-} hadrons$ and $\gamma^*\gamma^* \rightarrow hadrons$ processes are measured as a function of the photon virtualities, $Q^2_1$ and $Q^2_2$, of the two-photon mass, $W_{\gamma\gamma}$, and of the variable $Y= \ln ( W^2_{\gamma\gamma}/{Q_1 Q_2})$, for an average photon virtuality $\langle Q^2 \rangle = 16~\mathrm{GeV^2}$. The results are in agreement with next-to-leading order calculations for the process $\gamma^* \gamma^* \rightarrow \mathrm{q \bar{q}}$ in the interval $2 \le Y \le 5$. An excess is observed in the interval $5 < Y \le 7$, corresponding to $W_{\gamma\gamma}$ greater than 40~GeV . This may be interpreted as a sign of resolved photon QCD processes or the onset of BFKL phenomena.