  The contribution of the CP violating three-gluon Weinberg operator,
$\frac{1}{3!} w f^{abc} \epsilon^{\nu \rho \alpha \beta} G^a_{\mu \nu}
G^b_{\alpha \beta} G^{c \mu}_{\rho}$, to the atomic and nuclear EDMs is
estimated using QCD sum rules. After calculating the transition matrix element
between the pion and the vacuum through the Weinberg operator, we obtain the
long-range CP-odd nuclear force by determining the isovector CP-odd
pion-nucleon vertex, using chiral perturbation theory at NLO. The EDMs of
$^{199}$Hg, $^{129}$Xe, $^{225}$Ra, $^2$H, and $^3$He are finally given
including comprehensive uncertainty analysis. While the leading contribution of
the $^{199}$Hg EDM is given by the intrinsic nucleon EDM, that of $^{129}$Xe
atom may be dominated by the one-pion exchange CP-odd nuclear force generated
by the Weinberg operator. From current experimental data of the $^{199}$Hg
atomic EDM, we obtain an upper limit on the Weinberg operator magnitude of
$\left|w \right| < 4 \times 10^{-10} {\rm GeV}^{-2}$ if we assume that it is
the only source of CP violation at the scale $\mu =1$ TeV.
