We have measured the proper motion of the Large Magellanic Cloud (LMC) using 21 plates taken with the CTIO 4 m telescope and covering an epoch span of 14 yr. The plates were centered on the globular cluster NGC 2257, lying on the northeast periphery of the Cloud. Proper motions were determined for 251 LMC members, chosen on the basis of the photometry of Stryker [ApJS, 55, 127 (1984)], using 92 galaxies as a reference frame. The measured mean absolute proper motion of the LMC stars in our region is mu(alpha) = O.120'' +/- 0.028'' century-1, mu(delta) = 0.026'' +/- 0.027'' century-1. After correcting for the rotation of the LMC and the effects of solar motion, this proper motion combined with the radial velocity of the LMC implies a galactocentric coordinate radial velocity for the Cloud of 48 +/- 41 km s-1 and a total galactocentric transverse velocity of 215 +/- 48 km s-1.