We are making precise determinations of the abundance of the light isotope
of helium, He-3. The He-3 abundance in Milky Way sources provides strong co
nstraints for theories of stellar evolution, chemical evolution, and cosmol
ogy. Here we report on new He-3(+)/H+ abundance determinations for a sample
of 21 Galactic H II regions observed for the past 14 yr with the NRAO 140
Foot telescope. The abundances are based on improved He-3(+) observations,
as well as new models for source density structure. The models are constrai
ned by the observed He-3(+) emission, the strength of the H and He-4 recomb
ination lines, and the continuum emission at 3.5 cm wavelength. Density str
ucture serves only to increase the actual He-3(+)/H+ abundance ratio over t
hat derived using a uniform density sphere assumption. The density structur
e models allow one to identify sources that are "simple" in that their stru
cture corrections are negligible. These objects provide a pool of H II regi
ons that can in principle yield accurate He-3(+)/H+ abundances. For the sev
en simple H II regions in our sample there is no evidence for any systemati
c trend of the He-3(+)/H+ abundance with any other parameter. For these sou
rces, we derive an average abundance ratio by number of He-3(+)/H+ = (1.6 /- 0.5) x 10(-5) (statistical error).