  We carry out a high-precision simulation of the two-dimensional $SU(3)$
principal chiral model at correlation lengths $\xi$ up to $\approx\! 4 \times
10^5$, using a multi-grid Monte Carlo (MGMC) algorithm. We extrapolate the
finite-volume Monte Carlo data to infinite volume using finite-size-scaling
theory, and we discuss carefully the systematic and statistical errors in this
extrapolation. We then compare the extrapolated data to the
renormalization-group predictions. For $\xi \gtapprox 10^3$ we observe good
asymptotic scaling in the bare coupling; at $\xi \approx 4 \times 10^5$ the
nonperturbative constant is within 2--3\% of its predicted limiting value.
