We present the first results of a ground-based program to determine the proper motion of the Magellanic Clouds (MCs) relative to background quasars (QSO), being carried out using the Iréneé du Pont 2.5 m telescope at Las Campanas Observatory, Chile. Eleven QSO fields have been targeted in the Small Magellanic Cloud (SMC) over a time base of six years, and with seven epochs of observation. One quasar field was targeted in the Large Magellanic Cloud (LMC), over a time base of five years, and with six epochs of observation. The shorter time base in the case of the LMC is compensated by the much larger amount of high-quality astrometry frames that could be secured for the LMC quasar field (124 frames), compared to the SMC fields (an average of roughly 45 frames). In this paper, we present final results for field Q0557-6713 in the LMC and field Q0036-7227 in the SMC. From field Q0557-6713, we have obtained a measured proper motion of μαcos δ = +1.95 ± 0.13 mas yr-1, μδ = +0.43 ± 0.18 mas yr-1 for the LMC. From field Q0036-7227, we have obtained a measured proper motion of μα cosδ = +0.95 ± 0.29 mas yr-1, μδ = -1.14 ± 0.18 mas yr-1 for the SMC. Although we went through the full procedure for another SMC field (QJ0036-7225), on account of unsolvable astrometric difficulties caused by blending of the QSO image, it was impossible to derive a reliable proper motion. Current model rotation curves for the plane of the LMC indicate that the rotational velocity (V rot) at the position of LMC field Q0557-6713 can be as low as 50 km s-1, or as high as 120 km s-1. A correction for perspective and rotation effects leads to a center of mass proper motion for the LMC of μα cosδ = +1.82 ± 0.13 mas yr-1, μδ = +0.39 ± 0.15 mas yr-1 (V rot = 50 km s-1), and to μα cosδ = +1.61 ± 0.13 mas yr-1, μδ = +0.60 ± 0.15 mas yr-1 (V rot = 120 km s-1). Assuming that the SMC has a disk-like central structure, but that it does not rotate, we obtain a center of mass proper motion for the SMC of μα cosδ = +1.03 ± 0.29 mas yr-1, μδ = -1.09 ± 0.18 mas yr-1. Our results are in reasonable agreement with most previous determinations of the proper motion of the MCs, including recent Hubble Space Telescope measurements. Complemented with published values of the radial velocity of the centers of the LMC and SMC, we have used our proper motions to derive the galactocentric (gc) velocity components of the MCs. For the LMC, we obtain V gc,t = +315 ± 20 km s-1, V gc,r = +86 ± 17 km s-1 (V rot = 50 km s-1), and V gc,t = +280 ± 24 km s-1, V gc,r = +94 ± 17 km s-1 (V rot = 120 km s-1). For the SMC, we obtain V gc,t = +258 ± 50 km s-1, V gc,r = +20 ± 44 km s-1. These velocities imply a relative velocity between the LMC and SMC of 84 ± 50 km s-1, for V rot,LMC = 50 km s-1, and 62 ± 63 km s-1 for V rot,LMC = 120 km s-1. Albeit our large errors, these values are not inconsistent with the standard assumption that the MCs are gravitationally bound to each other.