With no will to cross Republican: that's the US way (10)
I believe the answer is:
interstate
'the us way' is the definition.
(type of motorway in the US)
'with no will to cross republican' is the wordplay.
'with no will' becomes 'intestate' (intestate means not having written a will).
'to cross' means one lot of letters goes inside another.
'republican' becomes 'r' (abbreviation).
'intestate' placed around 'r' is 'INTERSTATE'.
'that's' acts as a link.
(Other definitions for interstate that I've seen before include "American motorway" , "between parts of US" , "road in US" , "highway" , "American way" .)