The Collaborative International Dictionary of English v.0.48:

Exarchate \Ex*ar"chate\, n. [LL. exarchatus, fr. L. exarchus:
   cf. F. exarchat.]
   The office or the province of an exarch. --Jer. Taylor.
   [1913 Webster]

WordNet (r) 3.0 (2006):

exarchate
    n 1: a diocese of the Eastern Orthodox Church [syn: eparchy,
         exarchate]