The evolution of primordial stars of initial masses between 5 and 10 M⊙ has been computedand analyzed in order to determine the nature of the remnants of massive intermediate–massprimordial stars and to check the influence of overshooting in their evolution. We have obtainedthe values for the limiting masses of Population III progenitor stars leading to carbon–oxygenand oxygen–neon compact cores. We have also obtained the limiting mass for which isolatedprimordial stars would lead to core–collapse supernovae after the end of the main central burningphases. Considering a moderate amount of overshooting, the mass thresholds at the ZAMS for theformation of carbon–oxygen and oxygen–neon degenerate cores shift to smaller values by about2 M⊙. As a by–product of our calculations, we have also obtained the structure and compositionprofiles of the resulting compact remnants. We find that the final fate of the considered stars couldnot be to become white dwarfs, as it is the case of objects of larger metallicity of analogous initialmasses. Instead, as we show by means of a synthetic code, they might end their lives as SNI1/2.