End of period in States (8)
I believe the answer is:
terminus
'end' is the definition.
(terminus is a kind of end)
'period in states' is the wordplay.
'period' becomes 'term' (term is a kind of period**).
'states' becomes 'us' (abbreviation for the United States).
'term'+'in'+'us'='TERMINUS'
'of' is the link.
(Other definitions for terminus that I've seen before include "Roman god of boundaries" , "Destination point" , "End - objective" , "End, extremity" , "End of bus/rail route" .)