In the case Herrera v. Collins, the defendant petitioned for habeas relief claiming new evidence showed he was innocent. Unfortunately, Texas law has a thirty day deadline (after initial sentencing) for filing an appeal based upon new evidence. After that month, you're out of luck. The Supreme Court denied Herrera's claim and upheld Texas law. They wrote "Federal habeas courts do not sit to correct errors of fact, but to ensure that individuals are not imprisoned in violation of the Constitution."
This is usually interpreted to mean that it is not unconstitutional to put an innocent man to death so long as he has received a fair trial.