Presently-available sources of free neutrons can allow an improvement in the discovery potential of a neutron-antineutron transition search by four orders of magnitude as compared to that of the most recent reactor-based search experiment performed at ILL in Grenoble. This would be equivalent to a characteristic neutron-antineutron transition time limit of >10{sup 10} seconds. With future dedicated neutron-source Facilities, with further progress in cold-neutron- moderator techniques, and with a vertical experiment layout, the discovery potential could ultimately be pushed by another factor of {approximately}100 corresponding to a characteristic transition time limit of {approximately}10{sup 11} seconds. Prospects for, and relative merits of, a neutron-antineutron oscillation search in intranuclear transitions are also discussed
