  We carry out a high-precision Monte Carlo simulation of the two-dimensional
$O(3)$-invariant $\sigma$-model at correlation lengths $\xi$ up to $\sim 10^5$.
Our work employs a new and powerful method for extrapolating finite-volume
Monte Carlo data to infinite volume, based on finite-size-scaling theory. We
discuss carefully the systematic and statistical errors in this extrapolation.
We then compare the extrapolated data to the renormalization-group predictions.
The deviation from asymptotic scaling, which is $\approx 25\%$ at $\xi \sim
10^2$, decreases to $\approx 4\%$ at $\xi \sim 10^5$.
