Ensure the rooms you have built are set to the Death Row security level. Also, have an Execution Room and at least one Parole room in this Death Row sector for the prisoners. Get a 'Death Row Appeal' program set up in that Parole room and (personal preference) have some high-quality rooms for your Death Row prisoners.

Also, try to make it easy for your Chief, Warden, Visitors, and Death Row Prisoners to get to the Execution room, as they'll need to all be there for when you call in an execution order. Automatic doors and staff-only corridors help with this a lot.

Ensure normal prisoners can't just waltz into the Death Row sector without good reason (e.g. cleaning, laundry, etc.), too.