We present the results of a medium depth ecliptic sky survey conducted with telescopes at the Mauna Kea and Cerro-Tololo Observatories. The survey revealed 15 new Kuiper Belt objects, and 2 Centaurs (objects orbiting in the vicinity of the gas giant planets), bringing the total number of Kuiper Belt objects to 32 and Centaurs to 6. We use the new objects to begin to constrain some of the properties of the Kuiper Belt and Centaur populations. The apparent width of the Kuiper Belt is 10 degrees full width at half maximum (FWHM) but, after correcting for the effects of observational selection, we find that the intrinsic width must be at least 30 degrees FWHM. The inferred number of objects in the 30-50 AU heliocentric distance range is 7X10(4) (diameters greater than or equal to 100 km). Of these, about 40% (approximate to 3X10(4)) are in or near the 3:2 mean-motion resonance with Neptune. Pluto, which also occupies this resonance, is now seen as the largest of a hitherto unknown family of dynamically similar, resonantly trapped objects. We find that the Centaurs have a sky-plane surface density that is approximate to 0.5 deg(-2) to m(R)=24.2. The total population with absolute magnitude H-g less than or equal to 9.5 (diameter approximately greater than or equal to 75 km if albedo approximate to 0.04) is N approximate to 2600, about an order of-magnitude greater than the corresponding number of large main-belt asteroids. We crudely estimate that 1.5 M(Earth) of material has been cycled from the Kuiper Belt through the Centaurs in the age of the solar system. (C) 1996 American Astronomical Society.