We compute and analyze the evolution of primordial stars of masses at the ZAMS between 5 M_sun and 10 M_sun, with and without overshooting. Our main goals are to determine the nature of the remnants of massive intermediate-mass primordial stars and to check the influence of overshooting in their evolution. Our calculations cover stellar evolution from the main sequence phase until the formation of the degenerate cores and the thermally pulsing phase. We have obtained the values for the limiting masses of Population III progenitor stars leading to carbon-oxygen and oxygen-neon compact cores. Moreover, we have also obtained the limiting mass for which isolated primordial stars would lead to core-collapse supernovae after the end of the main central burning phases. Considering a moderate amount of overshooting the mass thresholds at the ZAMS for the formation of carbon-oxygen and oxygen-neon degenerate cores shifts to smaller values by about 2 M_sun. As a by-product of our calculations, we have also obtained the structure and composition profiles of the resulting compact remnants. Opposite to what happens with solar metallicity objects, the final fate of primordial stars is not straightforward determined from the mass of the compact cores at the end of carbon burning. Instead, the small mass-loss rates typically associated to stellar winds of low metallicity stars might allow the growth of the resulting degenerate cores up to the Chandrasekhar mass, on time scales one or two orders of magnitude shorter than the time required to loose the envelope. This would lead to the formation of supernovae for initial masses as small as about 5 M_sun.
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