  We performed a search for the decay $K_L^0 \rightarrow 3\gamma$ with the
E391a detector at KEK. In the data accumulated in 2005, no event was observed
in the signal region. Based on the assumption of $K_L^0 \rightarrow 3\gamma$
proceeding via parity-violation, we obtained the single event sensitivity to be
$(3.23\pm0.14)\times10^{-8}$, and set an upper limit on the branching ratio to
be $7.4\times10^{-8}$ at the 90% confidence level. This is a factor of 3.2
improvement compared to the previous results. The results of $K_L^0 \rightarrow
3\gamma$ proceeding via parity-conservation were also presented in this paper.
