Final answer:
The State Election Commission is the body mandated by the Illinois Constitution to supervise the state's registration and election laws.
Step-by-step explanation:
The government body that is required by the Constitution to supervise the carrying out of the registration and election laws of Illinois is the State Election Commission. While the Secretary of State maintains records and performs duties related to elections, it is the State Election Commission that has the specific mandate to oversee the actual administration of registration and election laws. This aligns with the broader principle that states are mainly responsible for their own elections, although there is federal oversight to ensure adherence to Constitutional requirements and standards.