An old story that bears fruit (8)
I believe the answer is:
chestnut
This might be a double definition.
'an old story' is the first definition.
(I've seen this before)
'bears fruit' is the second definition.
'chestnut' can be an answer for 'fruit' (chestnut is a kind of fruit). I am unsure of the 'bears' bit.
'that' acts as a link.
Can you help me to learn more?
(Other definitions for chestnut that I've seen before include "''Underneath the spreading . . . . . . . . tree''" , "Laughably predictable" , "Wooden" , "not a cracker!" , "Item for stuffing" .)