Safe American Food Exports Act of 2023 or the SAFE Act of 2023 This bill allows the Department of Agriculture (USDA) to preemptively negotiate regional export ban agreements for known animal disease threats that apply only to areas affected by animal disease outbreaks to enable the continuation of exports from areas not affected by an outbreak. The Animal and Plant Health Inspection Service, the Food Safety and Inspection Service (FSIS), and the Foreign Agricultural Service, in consultation with the Office of the U.S. Trade Representative, are authorized to negotiate the regionalization, zoning, compartmentalization, and other trade agreements regarding outbreaks of known animal disease threats of trade significance with countries with export markets for livestock animals or animal products from the United States. In addition, USDA must promulgate a regulation requiring that, within three days of removing any language from the FSIS Import and Export Library, FSIS notify each state department of agriculture, each lead state agency for animal disease, and any state and national producer organizations representing impacted livestock producers. (The FSIS library provides country and product specific information for (1) meat and poultry exports, and (2) products eligible for import to the United States.)