Great Ape Conservation Reauthorization Amendments Act of 2013 - Amends the Great Ape Conservation Act of 2000 to authorize the Secretary of the Interior to award a multi-year grant to an eligible person to implement a great ape conservation project that the person demonstrates is an effective, long-term conservation strategy for great apes and their habitats.

Requires the Secretary to convene a panel of experts to identify the greatest needs and priorities for the conservation of great apes within a year of this Act's enactment and every five years thereafter (current law authorizes the Secretary to convene a panel to consider the greatest conservation needs every two years). Requires the panel to include representatives from foreign range states with expertise in great ape conservation and to consider relevant great ape conservation plans or strategies, including scientific research and findings related to: (1) conservation needs and priorities of great apes, (2) regional or species-specific action plans or strategies, (3) applicable strategies developed or initiated by the Secretary, and (4) any other applicable conservation plan or strategy.

Authorizes the Secretary to pay expenses of convening and facilitating meetings of the panel.

Authorizes appropriations for the Great Ape Conservation Fund for FY2014-FY2018.