  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 affine
algebra of type $G_2^{(1)}$. We first determine singular vectors and then
obtain a description of the associative algebra $A(L(k,0))$ using the singular
vectors. 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 $L(k,0)$-modules
from the category $\mathcal O$ and that such an $L(k,0)$-module is completely
reducible. Our result supports the conjecture made by Adamovi{\'c} and Milas in
\cite{AM}.

Representation Theory;Quantum Algebra