The Collaborative International Dictionary of English v.0.48:

Cardinalate \Car"di*nal*ate\, n. [Cf. F. cardinalat, LL.
   cardinalatus.]
   The office, rank, or dignity of a cardinal.
   [1913 Webster]

WordNet (r) 3.0 (2006):

cardinalate
    n 1: cardinals collectively