Source : Webster's Revised Unabridged Dictionary (1913)

Viscountship \Vis"count`ship\, Viscounty \Vis"count`y\, n. [F.
   vicomt['e].]
   The quality, rank, or office of a viscount.

Source : WordNet®

viscounty
     n 1: the dignity or rank or position of a viscount or viscountess
          [syn: {viscountcy}]
     2: the domain controlled by a viscount or viscountess