Torsion balance measurements of the difference in ratios of gravitational to inertial mass for different materials have been carried out, confirming to higher precision the null results obtained 60 years ago by Eötvös and assumed by Einstein as the Principle of Equivalence upon which the General Theory of Relativity is founded. If the parameter η( A, B) is defined as η(A, B)=[(M/m) A-(M/m B]/1/2[M/m) A+(M/m) B], where M and m represent the passive gravitational and inertial masses respectively of materials A and B, then the results from the most sensitive torsion balance used enable us to conclude with 95% confidence that |η(Au, Al)| < 3 × 10 -11. Stated more exactly, the various measurements of η, obtained from the gravitational acceleration toward the sun, gave a substantially Gaussian distribution with mean value η(Au, Al) = (1.3 ± 1.0) × 10 -11. The probable error quoted for the mean is based upon the observed scatter in results from individual data runs, assuming a Gaussian distribution. The importance of the Eötvös experiment to contemporary gravitational theories is discussed, and the earlier measurements of Eötvös and J. Renner are examined critically. The torsion balance and associated equipment used in the present experiment are described in detail, along with the considerations involved in their design. Methods of data analysis are also discussed extensively and tables of individual results are presented.