The Collaborative International Dictionary of English v.0.48:

Begrime \Be*grime"\, v. t. [imp. & p. p. Begrimed; p. pr. &
   vb. n. Begriming.]
   To soil with grime or dirt deeply impressed or rubbed in.
   [1913 Webster]

         Books falling to pieces and begrimed with dust.
                                                  --Macaulay.
   [1913 Webster]

WordNet (r) 3.0 (2006):

begrime
    v 1: make soiled, filthy, or dirty; "don't soil your clothes
         when you play outside!" [syn: dirty, soil, begrime,
         grime, colly, bemire] [ant: clean, make clean]

Moby Thesaurus II by Grady Ward, 1.0:

21 Moby Thesaurus words for "begrime":
   bemire, bemud, besmoke, besoil, dirt, dirty, dirty up, dust, foul,
   grime, mire, muck, muck up, muddy, slime, smirch, smoke, smooch,
   smudge, soot, tarnish