© 2017 American Physical Society. We searched for the bound state of the neutron-rich Λ-hypernucleus HΛ6, using the Li6(π-,K+)X double charge-exchange reaction at a π- beam momentum of 1.2GeV/c at J-PARC. A total of 1.4×1012π- was driven onto a Li6 target of 3.5-g/cm2 thickness. No event was observed below the bound threshold, i.e., the mass of HΛ4+2n, in the missing-mass spectrum of the Li6(π-,K+)X reaction in the 2<θπK<20 angular range. Furthermore, no event was found up to 2.8MeV/c2 above the bound threshold. We obtained the double-differential cross section spectra of the Li6(π-,K+)X reaction in the angular range of 2<θπK<14. An upper limit of 0.56 nb/sr (90% C.L.) was obtained for the production cross section of the HΛ6 hypernucleus bound state. In addition, not only the bound state region, but also the Λ continuum region and part of the Σ- quasifree production region of the Li6(π-,K+) reaction were obtained with high statistics. The present missing-mass spectrum will facilitate the investigation of the Σ - nucleus optical potential for Σ - He5 through spectrum shape analysis.