The relative polarization of the two photons emitted when a positron annihilates at rest has been re-investigated with high precision and a different method of data analysis. An experiment using a pair of ideal polarization analyzers to measure this relative polarization would be a special case of the general class of thought experiments discussed by Einstein, Podolsky, and Rosen (EPR). EPR argued from these thought experiments that a physical system can exist in a state with definite values for two non-commuting variables. Since quantum mechanics can not describe such a state, EPR called quantum mechanics "incomplete". But EPR believed a complete theory -sometimes called a hidden variable theory- is possible. (This argument of EPR is sometimes called the Einstein-Podolsky-Rosen "paradox".) Our experimental results, together with a theorem due to Bell, provide strong evidence that a local "hidden variable" theory is not possible. The results also rule out a hypothetical modification of quantum mechanics, suggested by Bohm and Aharonov, which was motivated by the EPR thought experiments. Compton scattering was used to analyze the linear polarization. But the theorem of Bell, mentioned above, applies to relatively "ideal" polarization measurements. Therefore, it was necessary to prove the existence, and find the explicit form of the function f relating Compton and ideal linear polarization measurements. The existence of f is shown here to follow from general principles of quantum mechanics, plus parity and angular momentum conservation; the explicit form of f is deduced from the Klein-Nishina equation. Experimental evidence is cited against the argument that f may be different in a local "hidden variable" theory.