We have made a new calculation of the cosmic-ray secondary positron spectrum using a diffusive halo model for Galactic cosmic-ray propagation. The code computes self-consistently the spectra of primary and secondary nucleons, primary electrons, and secondary positrons and electrons. The models are first adjusted to agree with the observed cosmic-ray Boron/Carbon ratio, and the interstellar proton and Helium spectra are then computed; these spectra are used to obtain the source function for the

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... econdary positrons/electrons which are finally propagated with the same model parameters. The primary electron spectrum is evaluated, again using the same model. Fragmentation and energy losses are computed using realistic distributions for the interstellar gas and radiation fields, and diffusive reacceleration is also incorporated. Our study includes a critical re-evaluation of the secondary decay calculation for positrons. The predicted positron fraction is in good agreement with the measurements up to 10 GeV, beyond which the observed flux is higher than that calculated. Since the positron fraction is now accurately measured in the 1-10 GeV range our primary electron spectrum should be a good estimate of the true interstellar spectrum in this range, of interest for gamma ray and solar modulation studies. We further show that a harder interstellar nucleon spectrum, similar to that suggested to explain EGRET diffuse Galactic gamma ray observations above 1 GeV, can reproduce the positron observations above 10 GeV without requiring a primary positron component.