This paper describes the discovery circumstances of the first asteroids ever observed outside Earth's atmosphere but within the neighborhood of the Earth-Moon system. Four natural objects with diameters in the range 5-50 m were detected during a search for Earth-approaching asteroids conducted each month at the 0.91 m Spacewatch Telescope of the University of Arizona at Kitt Peak. An additional 19 Earth approachers with sizes in the range 50 m-5 km have also been discovered. These observations determine the cumulative flux of asteroids near Earth as a function of absolute magnitude, a function which has never been determined from direct observation. For asteroids larger than approximately 100 m, a power-law dependence with exponent approximately 0.9 is observed, consistent with their evolution from the main-belt population. At approximately 10 m, however, the flux is more than two orders of magnitude greater than this power-law extrapolation. These smaller Earth approachers are a special population which may be evolved from short-period comets, or which may be fragments from recent collisions among larger asteroids with perihelia near Earth.