Accountability Through Electronic Verification Act This bill amends the Illegal Immigration Reform and Immigrant Responsibility Act of 1996 to make the E-Verify program permanent. The bill requires: (1) federal agencies, contractors, and critical employers to participate in E-Verify; (2) all U.S. employers to participate in E-Verify within one year of enactment of this Act; and (3) employers using a contract, subcontract, or exchange to obtain labor to certify that they utilize E-Verify. The Department of Homeland Security (DHS) shall require the E-Verify participation of an employer or class of employers if DHS has reasonable cause to believe that the employer is or has been in material violation of the employment eligibility verification process under the Immigration and Nationality Act (INA). An employer’s failure to use E-Verify shall be treated as a violation of the INA requirement to verify employment eligibility and creates a rebuttable presumption that the employer knowingly hired, recruited, or referred an illegal alien. The bill: (1) increases civil and criminal penalties for specified hiring-related violations, and (2) establishes a good faith civil penalty exemption/reduction for certain hiring-related violations. Employers who are convicted of specified hiring related crimes or who have repeatedly committed specified hiring-related violations may be disbarred from federal contract, grant, or cooperative agreement participation. State and local governments may not prohibit employers from using E-Verify to determine the employment eligibility of new hires or current employees. The bill authorizes the verification of individuals before they are hired, recruited, or referred if the individual so consents. Employers are required to: (1) use E-Verify to verify the identity and employment eligibility of any individual who has not been previously verified through E-Verify not later than three years after enactment of this Act, (2) reverify the work authorization of individuals not later than three days after their employment authorization is due to expire, and (3) terminate an employee following receipt of a final E-Verify nonconfirmation and submit information DHS determines would assist it in enforcing or administering U.S. immigration laws. U.S. Citizenship and Immigration Services (USCIS) must report weekly to Immigration and Customs Enforcement regarding each person receiving a final E-Verify nonconfirmation. The Social Security Administration, DHS, and the Department of the Treasury shall establish an information sharing program. The bill: (1) provides for elimination of the Form I-9 process, and (2) sets forth E-Verify design and operation requirements. The federal criminal code is amended to: (1) provide that illegal aliens possessing or otherwise using false identification information not their own can be punished for identity fraud, and (2) subject a person who uses false identity information in furtherance of harboring or hiring illegal aliens to a fine and/or penalty of up to 20 years in prison. USCIS is required to establish a demonstration program to assist small businesses in rural areas or areas without Internet capabilities in verifying employment eligibility through the use of publicly accessible Internet terminals.