  The contribution of the CP violating three-gluon interaction, proposed by
Weinberg, to the short-range CP-odd nuclear force is evaluated in the
nonrelativistic quark model. We first show that the naive leading contribution
generated by the quark exchange process vanishes at sufficiently short distance
within the resonating group method, by considering the one-loop level gluon
exchange CP-odd interquark potential induced by the Weinberg operator with
massive quarks and gluons. We then estimate the true leading contribution by
evaluating the gluonic correction to the CP-odd interquark potential in the
closure approximation. It is found that the resulting irreducible CP-odd
nuclear force is comparable to that generated by the chiral rotation of the
CP-even short-range nuclear force, where the CP-odd mass calculated with QCD
sum rules is used as input. The explicit calculation of the electric dipole
moment (EDM) of the $^3$He nucleus yields $d_{^3{\rm He}}^{\rm (irr)} (w) = -
1.5 \, w\, e $ MeV. The total $^3$He EDM, accounting for the intrinsic nucleon
EDM, the pion-exchange and the short-range CP-odd nuclear force, is $d_{^3{\rm
He}}^{\rm (tot)} (w) = 20_{-11}^{+14} \, w\, e $ MeV, with the dominant effect
coming from the intrinsic nucleon EDM.
