The results of an extended examination of r-process nucleosynthesis in helium-burning enviroments are presented. Using newly calculated nuclear rates, dynamical r-process calculations have been made of thermal runaways in helium cores typical of low-mass stars and in the helium zones of stars undergoing supernova explosions. These calculations show that, for a sufficient flux of neutrons produced by the C-13 neutron source, r-process nuclei in solar proportions can be produced. The conditions required for r-process production are found to be 10 to the 20th-10 to the 21st neutrons per cubic centimeter for times of 0.01-0.1 s and neutron number densities in excess of 10 to the 19th per cubic centimeter for times of about 1 s. The amount of C-13 required is found to be exceedingly high - larger than is found to occur in any current stellar evolutionary model. It is thus unlikely that these helium-burning environments are responsible for producing the bulk of the r-process elements seen in the solar system.