The Collaborative International Dictionary of English v.0.48:

Undermine \Un`der*mine"\, v. t.
   1. To excavate the earth beneath, or the part of, especially
      for the purpose of causing to fall or be overthrown; to
      form a mine under; to sap; as, to undermine a wall.
      [1913 Webster]

            A vast rock undermined from one end to the other,
            and a highway running through it.     --Addison.
      [1913 Webster]

   2. Fig.: To remove the foundation or support of by
      clandestine means; to ruin in an underhand way; as, to
      undermine reputation; to undermine the constitution of the
      state.
      [1913 Webster]

            He should be warned who are like to undermine him.
                                                  --Locke.
      [1913 Webster]

WordNet (r) 3.0 (2006):

undermine
    v 1: destroy property or hinder normal operations; "The
         Resistance sabotaged railroad operations during the war"
         [syn: sabotage, undermine, countermine, counteract,
         subvert, weaken]
    2: hollow out as if making a cave or opening; "The river was
       caving the banks" [syn: cave, undermine]

Moby Thesaurus II by Grady Ward, 1.0:

88 Moby Thesaurus words for "undermine":
   abate, answer, answer conclusively, argue down, attenuate, blunt,
   bugger up, confound, confute, contradict, controvert, cramp,
   cripple, crush, damage, damp, dampen, dash, deaden, debilitate,
   defeat, demolish, deny, devitalize, disable, dismiss, dispose of,
   drain, dull, enervate, enfeeble, eviscerate, exhaust, extenuate,
   finish, floor, foil, frustrate, gruel, harm, honeycomb, hurt,
   impair, lay low, mine, mitigate, nonplus, overset, overthrow,
   overturn, overwhelm, parry, put to silence, queer, rattle, rebut,
   reduce, reduce to silence, refute, ruin, sabotage, sap,
   sap the foundations, settle, shake, shake up, shut up, silence,
   smash all opposition, soften up, spoil, squash, squelch, subvert,
   threaten, throw down, throw over, thwart, unbrace, undercut, unman,
   unnerve, unstrengthen, unstring, upend, upset, weaken, wreck