  This Letter provides a new determination of the ten real coefficients that
describe the structure of the $K \to \pi\pi\pi$ amplitudes in the limit where
isospin is conserved and complex phases, due to either CP violation or
final-state rescattering, are neglected. This determination is obtained through
a fit to the data on the Dalitz-plot structures and partial-decay rates
collected during the last twenty years by several high-precision experiments.
The fitting procedure and the way the experimental data have been handled in
the fit are discussed in detail. Our fit leads to a more precise determination
of the coefficients describing the linear and quadratic slopes of the $K \to
\pi\pi\pi$ amplitudes.
