The 15th Amendment granted the African American men the right to vote.
"Rights of the citizens of the United Sates to vote shall not be denied by the United States or bye any state on account of race, colour or prior condition of servitude."
The Southern States were able to disenfranchise the African Americans by presenting literacy tests, poll taxes and other means.
Although the amendment was passed on February 3rd 1870, it was over a century before this particular amendment to be recognised and it took the passage of the Voting Rights Act. of 1955 before the majority of African Americans living in the Southern States were even registered to vote.