Oral arguments will be heard April 2 in Atlanta before the Supreme Court. Melton recused himself without explanation. Weaver has been chief judge of the Appalachian Judicial Circuit since 1996.
Jones pleaded guilty to a quadruple slaying in Gordon County in January 2004. The victims were his infant daughter, his former girlfriend's parents and her sister.
The state is seeking the death penalty.
Among the issues before the high court are the trial court's decision to reject Jones' motion seeking to supress evidence found at his home and the denial of his motion seeking to bar imposition of the death penalty.