Final answer:
The Federal law that gives the FBI authority to investigate all federal crimes not assigned exclusively to another federal agency is the General Crimes Act.
Step-by-step explanation:
The Federal law that gives the FBI authority to investigate all federal crimes not assigned exclusively to another federal agency is the General Crimes Act. This law, also known as the Federal Major Crimes Act, allows the FBI to investigate a wide range of federal offenses, such as bank robberies, kidnappings, and drug trafficking, that are not under the jurisdiction of other federal agencies like the DEA or ATF.