We have identified and we discuss a variant of the r-process which occurs in a degenerate helium gas containing about 2% or more of /sup 13/C(..cap alpha..,n)/sup 16/O reaction serving as the energy and neutron production source. Thermal runaway leads to a slowly rising neutron flux which causes a buildup of heavier elements from lighter ones; in the final stages of the thermal flash the nuclei are transformed into an r-process distribution. It is of particular interest to know whether the solar system r-process distribution can be produced starting with negligible amounts of heavier elements at the beginning of the runaway. We have found that a series oif local helium flashes, followed by mixing throughout the helium region, conceptually a generalization of the Cole and Deupree helium flash process, leads to this results. Single helium flashes also lead to this result with 3% or more of /sup 13/C. The fit to the details of the solar system r-process abundance distribution is not good using r-process ..beta..-decay rates calculated by the gross theory, but if these rates are accelerated by amounts comparable to those advocated by Klapdor and his colleagues, a better fit is obtained.