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In this paper, we study representations of the vertex operator algebra$L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affinealgebra of type $G_2^{(1)}$. We first determine singular vectors and thenobtain a description of the associative algebra $A(L(k,0))$ using the singularvectors. We then prove that there are only finitely many irreducible$A(L(k,0))$-modules from the category $\mathcal O$. Applying the $A(V)$-theory,we prove that there are only finitely many irreducible weak $L(k,0)$-modulesfrom the category $\mathcal O$ and that such an $L(k,0)$-module is completelyreducible. Our result supports the conjecture made by Adamovi{\'c} and Milas in\cite{AM}.

Vertex Operator Algebras Associated to Type G Affine Lie Algebras