The Collaborative International Dictionary of English v.0.48:

Primate \Pri"mate\, n. [OE. primat, F. primat, L. primas, -atis
   one of the first, chief, fr. primus the first. See Prime,
   a.]
   1. The chief ecclesiastic in a national church; one who
      presides over other bishops in a province; an archbishop.
      [1913 Webster]

   2. (Zool.) One of the Primates.
      [1913 Webster]

WordNet (r) 3.0 (2006):

primate
    n 1: a senior clergyman and dignitary [syn: archpriest,
         hierarch, high priest, prelate, primate]
    2: any placental mammal of the order Primates; has good eyesight
       and flexible hands and feet

The Devil's Dictionary (1881-1906):

PRIMATE, n.  The head of a church, especially a State church supported
by involuntary contributions.  The Primate of England is the
Archbishop of Canterbury, an amiable old gentleman, who occupies
Lambeth Palace when living and Westminster Abbey when dead.  He is
commonly dead.