The Collaborative International Dictionary of English v.0.48:

Mesmerize \Mes"mer*ize\, v. t. [imp. & p. p. Mesmerized; p.
   pr. & vb. n. Mesmerizing.] [Also spelled mesmerise.]
   1. To bring into a state of mesmeric sleep; to hypnotize.
      [1913 Webster]

   2. To produce an intense fascination in; to spellbind.
      [PJC]

WordNet (r) 3.0 (2006):

mesmerise
    v 1: attract strongly, as if with a magnet; "She magnetized the
         audience with her tricks" [syn: magnetize, mesmerize,
         mesmerise, magnetise, bewitch, spellbind]
    2: induce hypnosis in [syn: hypnotize, hypnotise,
       mesmerize, mesmerise]