Promised to be busy (7)
I believe the answer is:
engaged
I believe this clue is a double definition.
'promised' is the first definition.
('engaged' can be a synonym of 'promised')
'be busy' is the second definition.
('engaged' can be similar in meaning to 'busy')
'to' is the link.
(Other definitions for engaged that I've seen before include "Hired; busy" , "Taken on - betrothed" , "Started to fight" , "May as well ring off" , "Plighted; hired" .)