  We study the running of the L\"uscher-Weisz-Wolff (LWW) coupling constant in
the two dimensional O(3) nonlinear $\sigma$ model. To investigate the continuum
limit we refine the lattice spacing from the $1\over 16$ value used by LWW up
to $1\over 160$. We find that the lattice artefacts are much larger than
estimated by LWW and that most likely the coupling constant runs slower than
predicted by perturbation theory. A precise determination of the running in the
continuum limit would require a controlled ansatz of extrapolation, which, we
argue, is not presently available.
