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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-volumeMonte Carlo data to infinite volume, based on finite-size-scaling theory. Wediscuss 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 $\xi \sim10^2$, decreases to $\approx 4\%$ at $\xi \sim 10^5$.Comment: 11 pages including 3 figures, 278964 bytes Postscript (changed  storage format

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Asymptotic Scaling in the Two-Dimensional O(3) σ Model at Correlation Length 1<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>

Asymptotic Scaling in the Two-Dimensional O(3) σ Model at Correlation Length 105