Firearm Recidivist Sentencing Act of 2013 - Amends the federal criminal code, with respect to additional penalties for using or carrying a firearm in the commission of a crime of violence or drug trafficking crime, to: (1) modify the penalty to require an additional prison term of not more than 5 years, not more than 7 years if the firearm is brandished, and not more than 10 years if the firearm is discharged (currently, not less than 5, 7, or 10 years respectively); (2) allow a prison term for any term of years (currently, minimum mandatory term of 25 years) for a person who has a prior conviction that has become final; (3) require an additional prison term of not more than 15 years (currently, not less than 15 years) for using, carrying, or possessing armor piercing ammunition during the commission of a crime of violence or drug trafficking crime; and (4) eliminate the death penalty for cases in which the use of such ammunition results in a death.