A grave old doctor, one of two ending conflict (4,4)
I believe the answer is:
atom bomb
'one of two ending conflict' is the definition.
(two atom bombs ended World War Two)
'a grave old doctor' is the wordplay.
'grave' becomes 'tomb' (synonyms).
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'doctor' becomes 'MB' (Bachelor of Medicine, Medicinae Baccalaureus).
'a'+'tomb'+'o'+'mb'='ATOM-BOMB'
(Other definitions for atom bomb that I've seen before include "Nuclear device" , "The first one fell on Hiroshima" , "Nuclear weapon" , "can be a big killer" , "Explosive device" .)