Communities United with Religious leaders for the Elimination of HIV/AIDS Act of 2013 or CURE Act of 2013 - Authorizes the Secretary of Health and Human Services (HHS), acting through the Deputy Assistant Secretary for Minority Health, to make grants to eligible health entities (i.e., a public health agency, a health center, including a center operated by an Indian tribe or organization, a community-based organization, or a faith-based organization) for: (1) HIV/AIDS education and outreach activities, (2) HIV/AIDS prevention and access to treatment, and (3) HIV/AIDS testing activities. Grants priority to grant applications that serve one or more minority groups (i.e., American Indians [including Alaska Natives, Eskimos, and Aleuts], Asian Americans, Native Hawaiians and other Pacific Islanders, Blacks, and Hispanics) with a specified HIV occurrence rate.

Authorizes the Secretary, acting through the Administrator of the Substance Abuse and Mental Health Services Administration, to make grants to eligible health entities to provide HIV testing, counseling, and referral for medical treatment to youth who are members of minority groups, are not more than age 18, are HIV positive or at risk for HIV/AIDS, including young men of racial minorities who have sex with men, and are engaged in substance abuse.

Authorizes the Secretary, acting through the Director of the Centers for Disease Control and Prevention (CDC), to make grants to eligible health entities for public health intervention and prevention activities with respect to one or more minority groups, including: (1) rapid HIV testing, (2) measures and activities to prevent the spread of HIV/AIDS and minimize its symptoms, (3) outreach activities targeting both females and males, and (4) referrals to health and mental health resources and health organizations.

Directs the Secretary, acting through the Director of the CDC, to: (1) expand and intensify HIV/AIDS prevention and education activities for minority groups, and (2) implement a national media outreach campaign that urges sexually active individuals who are members of minority groups to be tested for and know their HIV/AIDS status.

Authorizes the Secretary, acting through the Director of the National Center on Minority Health and Health Disparities, to make grants to public or private organizations with one or more published studies on behaviors to study behavioral factors that lead to increased HIV/AIDS prevalence in minority groups.