Carrying out of extreme sentence (9)
I believe the answer is:
execution
I believe this is a double definition.
'carrying out' is the first definition.
(synonyms)
'extreme sentence' is the second definition.
(sentence of death)
'of' acts as a link.
(Other definitions for execution that I've seen before include "The carrying out of" , "Topping" , "Eg, beheading" , "style of musical performance" , "Queen of Hearts' order" .)