You ceased to have any rights when you decided to kill selfishly for your own personal gain in the first place. You know the punishment - if you can't do the time then don't do the crime.
In Kent in the UK recently there was a murderer who has been released early from prison - for good behaviour whilst inside. He killed a young kid. This kids parents run the risk of bumping into their child's murderer in the high street every weekend when they go into town to go shopping. Nowadays, in this country, you really can get away with murder!
The point is that it is unconstitutional in the USA, not the UK.
If you commit a crime you may be sent to jail, and some of your rights are removed (ex. the right to bear arms), but, when you are given the death penalty, no rights remain for you to excercise.
Remember the pledge of allegiance? The last line goes: "... indivisible, with LIBERTY and justice FOR ALL."