Beak welcomes ordinary method of execution (5)
I believe the answer is:
noose
'method of execution' is the definition.
(I've seen this before)
'beak welcomes ordinary' is the wordplay.
'beak' becomes 'nose' (I've seen this before).
'welcomes' indicates putting letters inside (in sense of 'invites in').
'ordinary' becomes 'O' (abbreviation eg in O-Level, Ordinary Level).
'nose' placed around 'o' is 'NOOSE'.
(Other definitions for noose that I've seen before include "Fatal device" , "Loop with running knot" , "Loop - snare" , "Cord snare" , "Hangman's halter" .)