  We have measured the temperature dependence of the conductivity $\sigma_{xx}$
of a two-dimensional electron system deep into the localized regime of the
quantum Hall plateau transition. Using variable-range hopping theory we are
able to extract directly the localization length $\xi$ from this experiment. We
use our results to study the scaling behavior of $\xi$ as a function of the
filling factor distance $|\delta \nu|$ to the critical point of the transition.
We find for all samples a power-law behavior $\xi\propto|\delta\nu|^{-\gamma}$
with a universal scaling exponent $\gamma = 2.3$ as proposed theoretically.
