  The effects of a new triple-$\alpha$ reaction rate (OKK rate) on the helium
flash of a helium accreting neutron star in a binary system have been
investigated. Since the ignition points determine the properties of a
thermonuclear flash of type I X-ray bursts, we examine the cases of different
accretion rates, $dM/dt (\dot{M})$, of helium from $3\times10^{-10} M_{\odot}
\rm yr^{-1}$ to $3\times10^{-8} M_{\odot} \rm yr^{-1}$, which could cover the
observed accretion rates. We find that for the cases of low accretion rates,
nuclear burnings are ignited at the helium layers of rather low densities. As a
consequence, helium deflagration would be triggered for all cases of lower
accretion rate than $\dot{M}\simeq 3\times10^{-8} M_{\odot} \rm yr^{-1}$. We
find that OKK rate could be barely consistent with the available observations
of the X-ray bursts on the helium accreting neutron star. However this
coincidence is found to depend on the properties of crustal heating and the
neutron star model.We suggest that OKK rate would be reduced by a factor of
$10^{2-3}$ for $10^8$ K in the range of the observational errors.
