The preliminary upper limit of 3 × 10-36 counts per second per Cl37 atom set by Davis, Harmer, and Hoffman in an experiment to detect solar neutrinos establishes a sensitive, nearly opacity independent, relationship between low energy nuclear cross section factors and an upper limit on the sun's initial helium abundance. Using currently quoted values for the relevant cross section factors and conventional assumptions regarding the sun's makeup, consistency between theoretical estimates for the counting rate and the Davis, Harmer, and Hoffman limit can be achieved only if the sun's initial helium abundance Y (He4 abundance by mass) is less than 16-20, surprisingly low when compared with values (Y = 25-30) which have previously been estimated for the sun and for other objects in the galaxy as well. A higher upper limit on the solar helium abundance is possible if one or more of the following is true: (1) One (or more) of the relevant nuclear cross section factors lies beyond commonly quoted limits; (2) One (or more) of the relevant neutrino absorption cross sections has been overestimated; (3) The sun did not condense out of a chemically homogeneous nebula and/or pass through a homogeneous phase; (4) The opacity near the solar center is such that the sun possesses a large convective core. (5) Spin-down or meridional currents mix or have mixed matter over the entire solar interior on a time scale short compared to the age of the sun; (6) The sun is considerably younger than 4 1 2 billion years; (7) There are large-scale magnetic fields in the sun with strength up to 109 gauss near the center; (8) The gravitational constant increases with time. A lower limit on Y in the neighborhood of 15-18 is set by the choice of 4 1 2 billion years for the sun's age. If, therefore, the eventual upper limit on the counting rate set by the Davis et al. experiment is reduced much below the preliminary limit, a clear discrepency with conventional theory will be established, regardless of external arguments for higher Y. © 1969.