  We explore application of the domain wall fermion formalism of lattice QCD to
calculate the $K\to\pi\pi$ decay amplitudes in terms of the $K\to\pi$ and $K\to
0$ hadronic matrix elements through relations derived in chiral perturbation
theory. Numerical simulations are carried out in quenched QCD using domain-wall
fermion action for quarks and an RG-improved gauge action for gluons on a
$16^3\times 32\times 16$ and $24^3\times 32\times 16$ lattice at $\beta=2.6$
corresponding to the lattice spacing $1/a\approx 2$GeV. Quark loop contractions
which appear in Penguin diagrams are calculated by the random noise method, and
the $\Delta I=1/2$ matrix elements which require subtractions with the quark
loop contractions are obtained with a statistical accuracy of about 10%. We
confirm the chiral properties required of the $K\to\pi$ matrix elements.
Matching the lattice matrix elements to those in the continuum at $\mu=1/a$
using the perturbative renormalization factor to one loop order, and running to
the scale $\mu=m_c=1.3$ GeV with the renormalization group for $N_f=3$ flavors,
we calculate all the matrix elements needed for the decay amplitudes. With
these matrix elements, the $\Delta I=3/2$ decay amplitude shows a good
agreement with experiment in the chiral limit. The $\Delta I=1/2$ amplitude, on
the other hand, is about 50--60% of the experimental one even after chiral
extrapolation. In view ofthe insufficient enhancement of the $\Delta I=1/2$
contribution, we employ the experimental values for the real parts of the decay
amplitudes in our calculation of $\epsilon'/\epsilon$. We find that the $\Delta
I=3/2$ contribution is larger than the $\Delta I=1/2$ contribution so that
$\epsilon'/\epsilon$ is negative and has a magnitude of order $10^{-4}$.
Possible reasons for these unsatisfactory results are discussed.
