We have obtained moderate resolution (R = few thousand) spectra of the Na I
lambda lambda 5890, 5896 (Na D) absorption line in a sample of 32 far-IR-b
right starburst galaxies. In 18 cases, the Na D line in the nucleus is prod
uced primarily by interstellar gas, while cool stars contribute significant
ly in the others. In 12 of the 18 "interstellar-dominated" cases the Na D l
ine is blueshifted by over 100 km s(-1) relative to the galaxy systemic vel
ocity (the "outflow sources"), while no case shows a net redshift of more t
han 100 km s(-1). The absorption-line profiles in these outflow sources spa
n the range from near the galaxy systemic velocity to a maximum blueshift o
f similar to 400-600 km s(-1). The outflow sources are galaxies systematica
lly viewed more nearly face-on than the others. We therefore argue that the
absorbing material consists of ambient interstellar material that has been
entrained and accelerated along the minor axis of the galaxy by a hot star
burst-driven superwind. The Na D lines are optically thick, but indirect ar
guments imply total hydrogen column densities of N-H similar to few x 10(21
) cm(-2). This implies that the superwind is expelling matter at a rate com
parable to the star formation rate. This outflowing material is evidently v
ery dusty: we find a strong correlation between the depth of the Na D profi
le and the line-of-sight reddening. Typical implied values are E(B-V) = 0.3
-1 over regions several-to-10 kpc in size. We briefly consider some of the
potential implications of these observations. The estimated terminal veloci
ties of superwinds inferred from the present data and extant X-ray data are
typically 400-800 km(-1), are independent of the galaxy rotation speed, an
d are comparable to (substantially exceed) the escape velocities for L-* (d
warf) galaxies. The resulting selective loss of metals from shallower poten
tial wells can establish the mass-metallicity relation in spheroids, produc
e the observed metallicity in the intracluster medium, and enrich a general
IGM to of order 10(-1) solar metallicity. If the outflowing dust grains ca
n survive their journey into the IGM, their effect on observations of cosmo
logically distant objects would be significant.