Abstract - A technique for testing the weak-equivalence principle is presented. This technique involves the measurement of differential accelerations between two test masses of different materials (e.g., aluminum and gold) free falling inside a 3-m-long cryostat dropped from a 40 km altitude balloon. The free-fall duration is 30 s for a non-propelled cryostat. The falling test masses are part of a high-sensitivity differential detector with a foreseeable sensitivity in detecting differential accelerations of about 1.5x10^ 1213 g/sqrt(Hz) (at the liquid-nitrogen temperature of 77 K) and 1.5x10^ 1214 g/sqrt(Hz) (at the liquid-helium temperature of 4 K). The detector is spun about a horizontal axis at a frequency of typically 1 Hz in order to modulate the gravity signal during free fall. The estimated accuracies, with 95% confidence level, in testing the weak-equivalence principle in a 30 s integration time are 5 parts in 10^14 at the temperature of liquid nitrogen and 5 parts in 10^15 at the temperature of liquid helium
