The Fifteenth Amendment was the initial move in awarding complete voting rights to African Americans. But, Jim Crow laws generated more constraints on these rights. With the passage of the Voting Rights Act, the Fifteenth Amendment was implemented as voting constraints were lifted.
Fifteenth Amendment:
The Fifteenth Amendment to the United States Constitution forbids the central and state jurisdictions from rejecting a voter the right to vote based on that citizen's "race, color, or former position of subjection".