  An experiment to search for the $K^-pp$ bound state was performed via the
in-flight $^3$He($K^-,n)$ reaction using 5.3 $\times$ $10^9$ kaons at 1 GeV/$c$
at the J-PARC hadron experimental facility. In the semi-inclusive neutron
missing-mass spectrum at $\theta_{n}^{lab}=0^\circ$, no significant peak was
observed in the region corresponding to $K^-pp$ binding energy larger than 80
MeV, where a bump structure has been reported in the $\Lambda p$ final state in
different reactions. Assuming the state to be isotropically decaying into
$\Lambda p$, mass-dependent upper limits on the production cross section were
determined to be 30--180, 70--250, and 100--270 $\mu$b/sr, for the natural
widths of 20, 60, and 100 MeV, respectively, at 95\% confidence level.
