  In this paper we report on the measurement of the rate of inclusive $\pi^0$
production induced by charged-current neutrino interactions in a C$_8$H$_8$
target at a mean energy of 1.3 GeV in the K2K near detector. Out of a sample of
11,606 charged current neutrino interactions, we select 479 $\pi^0$ events with
two reconstructed photons. We find that the cross section for the inclusive
$\pi^0$ production relative to the charged-current quasi-elastic cross section
is
$$\frac{\sigma_{CC\pi^0}}{\sigma_{CCQE}}=0.426\pm0.032(stat.)\pm0.035(syst.)$$
The energy dependent cross section ratio is also measured. The results are
consistent with previous experiments for exclusive channels on different
targets.
