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We report on submillimetre bolometer observations of the isolated neutron star RX J1856.5−3754 using the Large Apex Bolometer Camera bolometer array on the Atacama Pathfinder Experiment telescope. No cold dust continuum emission peak at the position of RX J1856.5−3754 was detected. The 3σ flux density upper limit of 5 mJy translates into a cold dust mass limit of a few earth masses. We use the new submillimetre limit, together with a previously obtained H-band limit, to constrain the presence of a gaseous, circumpulsar disc. Adopting a simple irradiated disc model, we obtain a mass accretion limit of Graphic and a maximum outer disc radius of ∼1014 cm. By examining the projected proper motion of RX J1856.5−3754, we speculate about a possible encounter of the neutron star with a dense fragment of the CrA molecular cloud a few thousand years ago.