The differences between the surface structure of the near side and the far side of the Moon have been topics of interest ever since photographs of the far side have been available. One recurrent hypothesis is that a large impact on the near side has deposited ejecta on the far side, resulting in thicker crust there. Specific proposals were made by P.H. Cadogan for the Gargantuan Basin and by E.A. Whitaker for the Procellarum Basin. Despite considerable effort, no consensus has been reached on the existence of these basins. The problem of searching for such a basin is one of finding its signature in a somewhat chaotic field of basin and crater impacts. The search requires a model of the topographic shape of an impact basin and its ejecta field. Such a model is described, based on elevation data of lunar basins collected by the Lidar instrument of the Clementine mission and crustal thickness data derived from tracking Clementine and other spacecraft. The parameters of the model are scaled according to the principles of dimensional analysis and isostatic compensation in the early Moon. The orbital dynamics of the ejecta and the curvature of the Moon are also taken into account. Using such a scaled model, a search for the best fit for a large basin led to identification of a basin whose cavity covers more than half the Moon, including the area of all of the impact basins visible on the near side. The center of this basin is at 22 degrees east longitude and 8.5 degrees north latitude and its average radius is approximately 3,160 km. It is a megabasin, a basin that contains other basins (the far side South Pole-Aitken Basin also qualifies for that designation). It has been called the Near Side Megabasin. Much of the material ejected from the basin escaped the Moon, but the remainder formed an ejecta blanket that covered all of the far side beyond the basin rim to a depth of from 6 to 30 km. Isostatic compensation reduced the depth relative to the mean surface to a range of 1 5 km, but the crustal thickness data reveals the full extent of the original ejecta. The elevation profile of the ejecta deposited on the far side, together with modification for subsequent impacts by known basins (especially the far side South Pole-Aitken Basin) matches the available topographic data to a high degree. The standard deviation of the residual elevations (after subtracting the model from the measured elevations) is about one-half of the standard deviation of the measured elevations. A section on implications discusses the relations of this giant basin to known variations in the composition, mineralogy, and elevations of different lunar terranes.