  The coupling constants of the order $p^2$ low-energy weak effective
lagrangian can be determined from the $K\to\pi$ and $K\to 0$ weak matrix
elements, choosing degenerate quark masses for the first of these. However, for
typical values of quark masses in Lattice QCD computations, next-to-leading
$O(p^4)$ corrections are too large to be ignored, and will need to be included
in future analyses. Here we provide the complete $O(p^4)$ expressions for these
matrix elements obtained from Chiral Perturbation Theory, valid for partially
quenched QCD with $N$ degenerate sea quarks. Quenched QCD corresponds to the
special case N=0. We also discuss the role of the $\eta'$ meson in some detail,
and we give numerical examples of the size of chiral logarithms.
