  We present our result for the $K\to\pi\pi$ decay amplitudes for both the
$\Delta I=1/2$ and $3/2$ processes with the improved Wilson fermion action.
Expanding on the earlier works by Bernard {\it et al.} and by Donini {\it et
al.}, we show that mixings with four-fermion operators with wrong chirality are
absent even for the Wilson fermion action for the parity odd process in both
channels due to CPS symmetry. Therefore, after subtraction of an effect from
the lower dimensional operator, a calculation of the decay amplitudes is
possible without complications from operators with wrong chirality, as for the
case with chirally symmetric lattice actions. As a first step to verify the
possibility of calculations with the Wilson fermion action, we consider the
decay amplitudes at an unphysical quark mass $m_K \sim 2 m_\pi$. Our
calculations are carried out with $N_f=2+1$ gauge configurations generated with
the Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson fermion
action at $a=0.091\,{\rm fm}$, $m_\pi=280\,{\rm MeV}$, and $m_K=580\,{\rm MeV}$
on a $32^3\times 64$ ($La=2.9\,{\rm fm}$) lattice. For the quark loops in the
penguin and disconnected contributions in the $I=0$ channel, the combined
hopping parameter expansion and truncated solver method work very well for
variance reduction. We obtain, for the first time with a Wilson-type fermion
action, that ${\rm Re}A_0 = 60(36) \times10^{ -8}\,{\rm GeV}$ and ${\rm Im}A_0
=-67(56) \times10^{-12}\,{\rm GeV}$ for a matching scale $q^* =1/a$. The
dependence on the matching scale $q^*$ for these values is weak.
