In order to make a detailed analysis of the s-process nucleosynthesis occurring in massive stars, we followed the evolution of a 25 M. star of Population I from the ZAMS up to central carbon depletion. The evolutionary calculations were computed with the latest version of the FRANEC code under canonical prescriptions, varying the C-12(alpha, gamma)O-16 reaction rate within the range of present undertainties. The physical and chemical information provided by these models was then used to calculate the s-processing during core He burning, with a complete nuclear network of 450 isotopes from He up to Po and 41 branching points. An updated set of n-capture cross sections and weak interaction rates was adopted, which includes a number of recent experimental and theoretical improvements. Solar system abundances were taken from Anders and Grevesse. The mean neutron density is found to vary with time reaching a maximum value of about 10(6) n cm-3. The final neutron exposure is tau = 0.206 mb-1. In the results, emphasis is given to the strong production factors of the s-only isotopes Ge-70, Se-76, Kr-80, Kr-82, Sr-86, Sr-85, and do their relative ratios; important contributions are also obtained for Fe-58, Cu-63,65 Ni-61,64 Zn-66,67,68, Ga-69-71, Ge-72,73, Se-78. For A less-than-or-equal-to 56, together with C-12 and O-16, consistent overabundances are found for several isotopes, among which Ne-22, Mg-25,26, S-36, Cl-37, K-40. Despite the fall of the distribution beyond A congruent-to 90, nonnegligible enhancements of the rare isotopes Gd-152 and Ta-180 are obtained. A preliminary comparison is made between our findings and the predictions by the classical s-theory, which foresees about the same exposure, but uses a much higher constant mean neutron density. We investigated the effects of present uncertainties in the C-12(alpha, gamma)O-16 and Ne-22(alpha, n)Mg-25 reaction rates and in the neutron cross sections of crucial isotopes. The possibility of a further s-processing during the evolutionary phases following core He burning is discussed. The results of this work form the basis for a critical discussion of the contribution from massive stars to the solar abundance of the s-isotopes that will be performed in Paper II.