  The renormalized coupling $\gr$ defined through the connected 4-point
function at zero external momentum in the non-linear O(3) sigma-model in two
dimensions, is computed in the continuum form factor bootstrap approach with
estimated error $\sim 0.3%$. New high precision data are presented for $\gr$ in
the lattice regularized theory with standard action for nearly thermodynamic
lattices $L/\xi\sim 7$ and correlation lengths $\xi$ up to $\sim 122$ and with
the fixed point action for correlation lengths up to $\sim 12$. The agreement
between the form factor and lattice results is within $\sim 1%$. We also
recompute the phase shifts at low energy by measuring the two-particle energies
at finite volume, a task which was previously performed by L\"uscher and Wolff
using the standard action, but this time using the fixed point action.
Excellent agreement with the Zamolodchikov S-matrix is found.
