Edge: border of hat, not cap (3)
I believe the answer is:
rim
'edge' is the definition.
(rim is a kind of edge)
'border of hat not cap' is the wordplay.
'border of hat' becomes 'brim'.
'not cap' means to remove the first letter (in a down clue, cap suggests the top letter).
'brim' with its first letter taken off is 'RIM'.
(Other definitions for rim that I've seen before include "Upper or outer edge" , "Raised edge of a round object" , "Edge, border" , "Curved edge" , "An edge, canyon or cup" .)