The Collaborative International Dictionary of English v.0.48:

Undertake \Un`der*take"\, v. t. [imp. Undertook; p. p.
   Undertaken; p. pr. & vb. n. Undertaking.] [Under + take.]
   1. To take upon one's self; to engage in; to enter upon; to
      take in hand; to begin to perform; to set about; to
      attempt.
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            To second, or oppose, or undertake
            The perilous attempt.                 --Milton.
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   2. Specifically, to take upon one's self solemnly or
      expressly; to lay one's self under obligation, or to enter
      into stipulations, to perform or to execute; to covenant;
      to contract.
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            I 'll undertake to land them on our coast. --Shak.
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   3. Hence, to guarantee; to promise; to affirm.
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            And he was not right fat, I undertake. --Dryden.
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            And those two counties I will undertake
            Your grace shall well and quietly enjoiy. --Shak.
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            I dare undertake they will not lose their labor.
                                                  --Woodward.
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   4. To assume, as a character. [Obs.] --Shak.
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   5. To engage with; to attack. [Obs.]
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            It is not fit your lordship should undertake every
            companion that you give offense to.   --Shak.
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   6. To have knowledge of; to hear. [Obs.] --Spenser.
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   7. To take or have the charge of. [Obs.] "Who undertakes you
      to your end." --Shak.
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            Keep well those that ye undertake.    --Chaucer.
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The Collaborative International Dictionary of English v.0.48:

Undertaking \Un`der*tak"ing\, n.
   1. The act of one who undertakes, or engages in, any project
      or business. --Hakluyt.
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   2. That which is undertaken; any business, work, or project
      which a person engages in, or attempts to perform; an
      enterprise.
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   3. Specifically, the business of an undertaker, or the
      management of funerals.
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   4. A promise or pledge; a guarantee. --A. Trollope.
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WordNet (r) 3.0 (2006):

undertaking
    n 1: any piece of work that is undertaken or attempted; "he
         prepared for great undertakings" [syn: undertaking,
         project, task, labor]
    2: the trade of a funeral director

Moby Thesaurus II by Grady Ward, 1.0:

123 Moby Thesaurus words for "undertaking":
   accomplished fact, accomplishment, achievement, act, acta, action,
   activities, activity, adventure, affair, affairs, agreement,
   approach, assay, assurance, attempt, bag, bail, bid, blow, bond,
   business, commerce, commitment, concern, concernment, contract,
   coup, crack, dealings, deed, doing, doings, earnest, earnest money,
   effort, employ, employment, endeavor, engagement, enterprise,
   escrow, essay, experiment, exploit, fait accompli, feat, fling,
   function, gage, gambit, gest, go, guarantee, guaranty, hand,
   handiwork, handsel, hassle, hock, hostage, interest, job, labor,
   lick, lookout, mainprise, maneuver, matter, measure, move,
   obligation, occupation, offer, operation, overt act, passage, pawn,
   performance, performing, pignus, pledge, preengagement, proceeding,
   production, project, promise, realization, recognizance, replevin,
   replevy, res gestae, service, shot, stab, step, striving, stroke,
   strong bid, struggle, stunt, surety, task, tentative, thing,
   thing done, token payment, tour de force, transaction, trial,
   trial and error, try, turn, understanding, vadimonium, vadium,
   venture, verbal agreement, vow, warranty, whack, work, works



Bouvier's Law Dictionary, Revised 6th Ed (1856):

UNDERTAKING, contracts. An engagement by one of the parties to a contract to
the other, and not the mutual engagement of the parties to each other; a
promise. 5 East, R. 17; 2 Leon. 224, 5; 4 B, & A. 595.