Accountability Through Electronic Verification Act This bill expands the E-Verify program by requiring all employers to use it, and permanently reauthorizes the program. Currently, E-Verify use is voluntary for most employers, although some states mandate its use. All employers shall use E-Verify to confirm the identity and employment eligibility of all recruited, referred, or hired individuals, including current employees who were never verified under the program. Failure to use E-Verify shall create a rebuttable presumption that the employer is violating immigration law. U.S. Citizenship and Immigration Services (USCIS) shall generate weekly reports about individuals who have received a final non-confirmation of employment eligibility. The Department of Homeland Security (DHS) shall use the report to enforce immigration laws. DHS shall establish a program to help certain small businesses verify employee eligibility. DHS shall also update E-Verify's design to help prevent and detect fraud and identity theft. The bill increases civil and criminal penalties for hiring unauthorized aliens. DHS shall debar repeat offenders and those criminally convicted from holding federal contracts, grants, or cooperative agreements. The Social Security Administration, Internal Revenue Service, Department of the Treasury, and DHS shall jointly establish a program to share information to help identify unauthorized aliens. The bill establishes the Employer Compliance Inspection Center within Homeland Security Investigations of U.S. Immigration and Customs Enforcement (ICE). The center's duties include processing I-9 employment eligibility verification forms and ensuring compliance with employment eligibility laws. DHS shall report to Congress on ways to simplify procedures relating to I-9 forms, and on whether the I-9 process should be eliminated.