Frank et al. [1986b> and Frank and Sigwarth [1993> hypothesized the intense bombardment of the terrestrial atmosphere by small comets. Their model requires that the Moon is impacted by small comets (107--108 g) at a rate of almost one per minute. We calculate that an object of this mass, even with an exceedingly low density and relatively low velocity, will nevertheless produce a crater at least 50 m in diameter. These craters will excavate immature lunar soil and produce a very bright spot with a diameter of at least 150 m. If low-density comets exist that might not create deep craters [O'Keefe and Ahrens, 1982>, they will nevertheless disturb the regolith sufficiently to create detectable bright spots. If the small-comet hypothesis is correct then the near-global lunar imaging returned by Clementine in 1994 should reveal ~107 bright spots in locations where craters are not present in images acquired in the 1960's and early 1970's. We find no new bright spots in a carefully-studied area of 5.2¿104 km2, so an upper limit to the current cratering rate by small comets is 33/yr, ~104 below that expected if the small-comet hypothesis were valid. ¿ 1997 American Geophysical Union