Edith Piaf's way to experience regret (3)
I believe the answer is:
rue
I believe this clue is a double definition.
'edith piaf's way' is the first definition.
(French for a road or way - Edith Piaf was French)
'experience regret' is the second definition.
(to rue something is to regret it)
'to' is the link.
(Other definitions for rue that I've seen before include "Bemoan" , "Cry" , "Regret the bitter herb" , "Wish not to have been" , "Regret, be sorry for" .)