We measure the seismic moment of five great (Ms?7.8) earthquakes from 1928 to 1942 along the Mexican portion of the Middle American Trench directly from recorded seismograms, and obtain considerably lower moment values than had previously been estimated from Ms. Such direct moment determinations are inherently more reliable. As the average seismic slip rate along this portion of the arc appears to be essentially constant, these measurements of the past seismic moment are used to estimate that the current (end of 1980) moment deficiency, is 46¿1027 dyne-cm, or an average slip of 97 cm along the arc. Although this is much lower than past estimates at the current rate of increase, 1.6¿1027 dyne-cm/yr (3.4 cm/yr of seismic displacement), the moment deficiency by 1985 will be the equivalent of five magnitude 8 earthquakes; and by 1990, six such events. The slip along this portion of the Middle America Trench are appears to fit Shimazaki and Nakata's ''slip predictable'' model, in which the size, but not the time of occurrence, of future large events may be esimated.