That the death penalty does not violate the Eighth Amendment to the Constitution was ruled in:

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The ruling that the death penalty does not violate the Eighth Amendment was established in the case of Gregg v. Georgia. In this case, the Supreme Court upheld a revised death penalty statute from Georgia that provided for bifurcated trials and automatic appeal. The Court found that the death penalty, when carefully administered under appropriate standards and procedures, does not constitute cruel and unusual punishment.

That the Death Penalty Does Not Violate the Eighth Amendment