The Collaborative International Dictionary of English v.0.48:

Demure \De*mure"\, v. i.
   To look demurely. [Obs.] --Shak.
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The Collaborative International Dictionary of English v.0.48:

Demure \De*mure"\ (d[-e]*m[=u]r"), a. [Perh. from OF. de murs
   (i. e., de bonnes murs of good manners); de of + murs, mours,
   meurs, mors, F. m[oe]urs, fr. L. mores (sing. mos) manners,
   morals (see Moral); or more prob. fr. OF. me["u]r, F.
   m[^u]r mature, ripe (see Mature) in a phrase preceded by
   de, as de m[^u]re conduite of mature conduct.]
   1. Of sober or serious mien; composed and decorous in
      bearing; of modest look; staid; grave.
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            Sober, steadfast, and demure.         --Milton.
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            Nan was very much delighted in her demure way, and
            that delight showed itself in her face and in her
            clear bright eyes.                    --W. Black.
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   2. Affectedly modest, decorous, or serious; making a show of
      gravity.
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            A cat lay, and looked so demure, as if there had
            been neither life nor soul in her.    --L'Estrange.
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            Miss Lizzy, I have no doubt, would be as demure and
            coquettish, as if ten winters more had gone over her
            head.                                 --Miss
                                                  Mitford.
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WordNet (r) 3.0 (2006):

demure
    adj 1: affectedly modest or shy especially in a playful or
           provocative way [syn: coy, demure, overmodest]