A balloon-borne instrument comprising a fast ionization chamber interposed in a G-M counter train has been flown near the geomagnetic equator for the purpose of determining the fluxes of protons and He nuclei in the primary cosmic radiation. The apparatus measures the specific ionization of each particle that traverses the telescope, and contains a shower detector to aid in the identification of multiple-particle events of either local or external origin. Under the assumption that all telescope coincidences resulted from particles incident within the counter-defined solid angle, the total vertical flux, extrapolated to the "top of the atmosphere," is Itotal=260 sterad-1 m-2 sec-1. The "top"-extrapolated fluxes of primary protons and He nuclei deduced from the ionization distributions recorded at various depths (17 g/cm2 and greater) are as follows: IH<145 sterad-1 m-2 sec-1; IHe<38 sterad-1 m-2 sec-1. The large difference between Itotal and (IH+IHe) is believed to arise from air showers and/or nuclear events produced by particles incident from outside the telescope solid angle. The consequences of neglecting these spurious effects in interpretating counter-telescope measurements are discussed with particular reference to influences on the deduced form of the primary-proton energy spectrum, and the total energy flux associated with the primary radiation.