  We report the first lattice QCD calculation of the complex kaon decay
amplitude $A_0$ with physical kinematics, using a $32^3\times 64$ lattice
volume and a single lattice spacing $a$, with $1/a= 1.3784(68)$ GeV. We find
Re$(A_0) = 4.66(1.00)(1.26) \times 10^{-7}$ GeV and Im$(A_0) =
-1.90(1.23)(1.08) \times 10^{-11}$ GeV, where the first error is statistical
and the second systematic. The first value is in approximate agreement with the
experimental result: Re$(A_0) = 3.3201(18) \times 10^{-7}$ GeV while the second
can be used to compute the direct CP violating ratio
Re$(\varepsilon'/\varepsilon)=1.38(5.15)(4.59)\times 10^{-4}$, which is
$2.1\sigma$ below the experimental value $16.6(2.3)\times 10^{-4}$. The real
part of $A_0$ is CP conserving and serves as a test of our method while the
result for Re$(\varepsilon'/\varepsilon)$ provides a new test of the
standard-model theory of CP violation, one which can be made more accurate with
increasing computer capability.
