Nuclear Weapons Abolition and Economic and Energy Conversion Act of 2013 - Requires the government: (1) to provide leadership to negotiate and enter into a multilateral treaty or other international agreement that provides for the dismantlement and elimination, under strict international control, of all nuclear weapons in every country by 2020; (2) once the President certifies that all countries have eliminated such weapons or begun such elimination under established legal requirements, to redirect resources that are being used for nuclear weapons programs to addressing human and infrastructure needs and to converting nuclear weapons industry employees, processes, plants, and programs to constructive, ecologically beneficial peacetime activities; (3) to undertake efforts to eliminate war, armed conflict, and all military operations; and (4) to promote policies to induce all other countries to join in such commitments.