Very sensitive H I 21 cm observations have been made in 860 directions at δ≥-43 &j0; in search of weak, Galactic, high-velocity H I emission lines at moderate and high Galactic latitudes. One-third of the observations were made toward extragalactic objects that are visible at optical and UV wavelengths. The median rms noise in the survey spectra is 3.4 mK, resulting in a median 4 σ detection level of N= 8× 10 17 cm-2 averaged over the 21 &arcmin; beam of the telescope. High-velocity H I emission is detected in 37% of the directions; about half of the lines could not have been detected in previous surveys. The median FWHM of detected lines is 30.3 km s-1. High-velocity H I lines are seen down to the sensitivity limit of the survey, implying that there are likely lines at still lower values of N. The weakest lines have a kinematics and distribution on the sky similar to that of the strong lines and thus do not appear to be a new population. Most of the emission originates from objects which are extended over several degrees; only a few appear to be compact sources. At least 75%, and possibly as many as 90%, of the lines are associated with one of the major high-velocity complexes. With the increased sensitivity of this survey, the Magellanic Stream is seen to extend at least 10 to higher Galactic latitude than previously thought and to be more extended in longitude as well. Wright's Cloud near M33 has an extended low-N component in the direction of the Magellanic Stream. The bright H I features which have dominated most surveys may be mere clumps within larger structures, and not independent objects. Although there are many lines with low column density, their numbers do not increase as rapidly as N, so most of the H I mass in the high-velocity cloud phenomenon likely resides in the more prominent clouds.