The Collaborative International Dictionary of English v.0.48:

Archbishopric \Arch`bish"op*ric\, n. [AS. arcebiscopr[imac]ce.
   See -ric.]
   The jurisdiction or office of an archbishop; the see or
   province over which archbishop exercises archiepiscopal
   authority.
   [1913 Webster]

WordNet (r) 3.0 (2006):

archbishopric
    n 1: the territorial jurisdiction of an archbishop

Moby Thesaurus II by Grady Ward, 1.0:

152 Moby Thesaurus words for "archbishopric":
   Kreis, abbacy, aedileship, archdeaconry, archdiocese,
   archiepiscopacy, archiepiscopate, aristocracy, arrondissement,
   bailiwick, bishopdom, bishopric, borough, canonicate, canonry,
   canton, cardinalship, chairmanship, chancellery, chancellorate,
   chancellorship, chaplaincy, chaplainship, chiefery, chiefry,
   chieftaincy, chieftainry, chieftainship, city, commune, conference,
   congressional district, constablewick, consulate, consulship,
   county, curacy, deaconry, deaconship, deanery, deanship,
   departement, dictatorship, dictature, diocese, directorship,
   district, duchy, electoral district, electorate, emirate,
   episcopacy, episcopate, government, governorship, hamlet, headship,
   hegemony, hierarchy, hundred, leadership, lordship, magistracy,
   magistrateship, magistrature, masterdom, mastership, mastery,
   mayoralty, mayorship, metropolis, metropolitan area,
   metropolitanate, metropolitanship, nobility, oblast, okrug, papacy,
   parish, pashadom, pashalic, pastorate, pastorship, patriarchate,
   patriarchy, pontificality, pontificate, popedom, popehood,
   popeship, prebend, prebendaryship, precinct, prefectship,
   prefecture, prelacy, prelateship, prelature, premiership,
   presbyterate, presbytery, presidency, presidentship, primacy,
   prime-ministership, prime-ministry, princedom, princeship,
   principality, proconsulate, proconsulship, protectorate,
   protectorship, province, provostry, provostship, rectorate,
   rectorship, regency, regentship, region, riding, ruling class, see,
   seigniory, seneschalship, seneschalsy, sheikhdom, sheriffalty,
   sheriffcy, sheriffdom, sheriffwick, shire, shrievalty, soke, stake,
   state, supervisorship, suzerainship, suzerainty, synod, territory,
   town, township, tribunate, vicariate, vicarship, village,
   vizierate, viziership, wapentake, ward