  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. In
order to realize the physical kinematics, where the pions in the final state
have finite momenta, we consider the decay process $K({\bf p}) \to \pi({\bf p})
+ \pi({\bf 0})$ in the nonzero momentum frame with momentum ${\bf
p}=(0,0,2\pi/L)$ on the lattice. 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 $a=0.091\,{\rm fm}$
($1/a=2.176\,{\rm GeV}$), $m_\pi=260\,{\rm MeV}$, and $m_K=570\,{\rm MeV}$ on a
$48^3\times 64$ ($La=4.4\,{\rm fm}$) lattice. For these parameters the energy
of the $K$ meson is set at that of two-pion in the final state. We obtain ${\rm
Re}A_2 = 2.431(19) \times10^{ -8}\,{\rm GeV}$, ${\rm Re}A_0 = 51(28) \times10^{
-8}\,{\rm GeV}$, and $\epsilon'/\epsilon = 1.9(5.7) \times10^{-3}$ for a
matching scale $q^* =1/a$ where the errors are statistical. The dependence on
the matching scale $q^*$ of these values is weak. The systematic error arising
from the renormalization factors is expected to be around $1.3\%$ for ${\rm
Re}A_2$ and $11 \%$ for ${\rm Re}A_0$. Prospects toward calculations with the
physical quark mass are discussed.
