A key aim of the HighNESS project for the European Spallation Source is to enable cutting-edge particle physics experiments. This volume presents a conceptual design report for the NNBAR experiment. NNBAR would exploit a new cold lower moderator to make the first search in over thirty years for free neutrons converting to anti-neutrons. The observation of such a baryon-number-violating signature would be of fundamental significance and tackle open questions in modern physics, including the origin of the matter-antimatter asymmetry. This report shows the design of the beamline, supermirror focusing system, magnetic and radiation shielding, and anti-neutron detector necessary for the experiment. A range of simulation programs are employed to quantify the performance of the experiment and show how background can be suppressed. For a search with full background suppression, a sensitivity improvement of three orders of magnitude is expected, as compared with the previous search. Civil engineering studies for the NNBAR beamline are also shown, as is a costing model for the experiment.|[GRAPHICS]|.