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We present high-resolution (0.5 km/s) spectra, obtained with the McDonald Observatory 2.7 m coude echelle spectrograph, of interstellar Na I D1 absorption toward 38 bright stars. Numerous narrow, closely blended absorption components, showing resolved Na I hyperfine structure, are evident in these spectra; such narrow components appear in both low halo and quite local gas, as well as in gas toward more distant disk stars. We have used the method of profile fitting in an attempt to determine column densities, line widths, and velocities for the individual interstellar clouds contributing to the observed absorption lines. The resulting sample of 276 clouds is significantly larger, and likely more complete, than several previous samples of 'individual' interstellar clouds, and allows more precise investigation of various statistical properties. We find that the cloud column density (N) and line width parameter (b) are not correlated, for 0.3 km/s approximately less than b approximately less than 1.5 km/s and 10.0/sq cm approximately less than log (N(Na I)) approximately less than 11.6/sq cm. The median b is about 0.73 km/s, the median log N is about 11.09/sq cm, and the median separation between adjacent components is about 2.0 km/s. All these are overestimates of the true median values, however, due to our inability to completely resolve all the component structure present in some cases; even at a resolution of 0.5 km/s, we may have discerned only 60% of the full number of individual components actually present. The one-dimensional dispersion of component velocities, in the local standard of rest, is approximately 8.6 km/s; the distribution of velocities is broader and displaced to more negative velocities for the weaker components. If 80 K is a representative temperature for the interstellar clouds seen in Na I absorption, then at least 38% (and probably the majority) of the clouds have subsonic internal turbulent motions. The range in N(H I) observed at a given N(Na I) increases as N(Na I) decreases below about 10<SUP>11</SUP>/sq cm, so that N(Na I) becomes a less reliable predictor of N(H I) at low column densities. These spectra will be quite useful in future detailed studies of these lines of sight with the Hubble Space Telescope (HST) Goddard High Resolution Spectrograph (GHRS) echelle -- to determine accurate velocities and to reveal the detailed interstellar component structures that cannot be discerned at the 3.5 km/s resolution available with the GHRS.