The Collaborative International Dictionary of English v.0.48:

Accomplish \Ac*com"plish\, v. t. [imp. & p. p. Accomplished,
   p. pr. & vb. n. Accomplishing.] [OE. acomplissen, OF.
   accomplir, F. accomplir; L. ad + complere to fill up,
   complete. See Complete, Finish.]
   1. To complete, as time or distance.
      [1913 Webster]

            That He would accomplish seventy years in the
            desolations of Jerusalem.             --Dan. ix. 2.
      [1913 Webster]

            He had accomplished half a league or more.
                                                  --Prescott.
      [1913 Webster]

   2. To bring to an issue of full success; to effect; to
      perform; to execute fully; to fulfill; as, to accomplish a
      design, an object, a promise.
      [1913 Webster]

            This that is written must yet be accomplished in me.
                                                  --Luke xxii.
                                                  37.
      [1913 Webster]

   3. To equip or furnish thoroughly; hence, to complete in
      acquirements; to render accomplished; to polish.
      [1913 Webster]

            The armorers accomplishing the knights. --Shak.
      [1913 Webster]

            It [the moon] is fully accomplished for all those
            ends to which Providence did appoint it. --Wilkins.
      [1913 Webster]

            These qualities . . . go to accomplish a perfect
            woman.                                --Cowden
                                                  Clarke.
      [1913 Webster]

   4. To gain; to obtain. [Obs.] --Shak.
      [1913 Webster]

   Syn: To do; perform; fulfill; realize; effect; effectuate;
        complete; consummate; execute; achieve; perfect; equip;
        furnish.

   Usage: To Accomplish, Effect, Execute, Achieve,
          Perform. These words agree in the general idea of
          carrying out to some end proposed. To accomplish (to
          fill up to the measure of the intention) generally
          implies perseverance and skill; as, to accomplish a
          plan proposed by one's self, an object, a design, an
          undertaking. "Thou shalt accomplish my desire." --1
          Kings v. 9.
          [1913 Webster]

                He . . . expressed his desire to see a union
                accomplished between England and Scotland.
                                                  --Macaulay.
          [1913 Webster] To effect (to work out) is much like
          accomplish. It usually implies some degree of
          difficulty contended with; as, he effected or
          accomplished what he intended, his purpose, but
          little. "What he decreed, he effected." --Milton.
          [1913 Webster]

                To work in close design by fraud or guile
                What force effected not.          --Milton.
          [1913 Webster] To execute (to follow out to the end,
          to carry out, or into effect) implies a set mode of
          operation; as, to execute the laws or the orders of
          another; to execute a work, a purpose, design, plan,
          project. To perform is much like to do, though less
          generally applied. It conveys a notion of protracted
          and methodical effort; as, to perform a mission, a
          part, a task, a work. "Thou canst best perform that
          office." --Milton.
          [1913 Webster]

                The Saints, like stars, around his seat
                Perform their courses still.      --Keble.
          [1913 Webster] To achieve (to come to the end or
          arrive at one's purpose) usually implies some
          enterprise or undertaking of importance, difficulty,
          and excellence.
          [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Accomplished \Ac*com"plished\, a.
   1. Completed; effected; established; as, an accomplished
      fact.
      [1913 Webster]

   2. Complete in acquirements as the result usually of
      training; -- commonly in a good sense; as, an accomplished
      scholar, an accomplished villain.
      [1913 Webster]

            They . . . show themselves accomplished bees.
                                                  --Holland.
      [1913 Webster]

            Daughter of God and man, accomplished Eve. --Milton.
      [1913 Webster]

WordNet (r) 3.0 (2006):

accomplished
    adj 1: highly skilled; "an accomplished pianist"; "a complete
           musician" [syn: accomplished, complete]
    2: successfully completed or brought to an end; "his mission
       accomplished he took a vacation"; "the completed project";
       "the joy of a realized ambition overcame him" [syn:
       accomplished, completed, realized, realised]
    3: settled securely and unconditionally; "that smoking causes
       health problems is an accomplished fact" [syn:
       accomplished, effected, established]