  A puzzle has long existed for the $\alpha$-cluster content in the
near-threshold 7.54 MeV state of $^{10}$Be. A new measurement was conducted to
measure the cluster-decay partial width of this state, using the reaction
$\rm{^9Be}(\rm{^9Be}, \rm{^{10}Be}^{*} \rightarrow \alpha +
\rm{^6He})\rm{^8Be}$ at 45 MeV beam energy. Special measures were taken to
reduce the strong near-threshold background. The neutron-decay strength was
also obtained based on the three-fold coincident measurement. A cluster-decay
branching ratio of $(4.04 \pm 1.26)\times 10^{-4}$ is obtained, resulting in a
reasonably large $\alpha$-cluster spectroscopic factor. The present work
confirms the formation of the $\sigma$-bond molecular rotational band headed by
the 6.18 MeV state in $^{10}$Be.
