Although they are but a small fraction of the mass ejected in core-collapse supernovae, neutrino-driven winds (NDWs) from nascent proto-neutron stars (PNSs) have the potential to contribute significantly to supernova nucleosynthesis. In previous works, the NDW has been implicated as a possible source of r-process and light p-process isotopes. In this paper, we present time-dependent hydrodynamic calculations of nucleosynthesis in the NDW which include accurate weak interaction physics coupled to a full nuclear reaction network. Using two published models of PNS neutrino luminosities, we predict the contribution of the NDW to the integrated nucleosynthetic yield of the entire supernova. For the neutrino luminosity histories considered, no true r-process occurs in the most basic scenario. The wind driven from an older 1.4 M{sub sun} model for a PNS is moderately neutron-rich at late times however, and produces {sup 87}Rb, {sup 88}Sr, {sup 89}Y, and {sup 90}Zr in near solar proportions relative to oxygen. The wind from a more recently studied 1.27 M{sub sun} PNS is proton-rich throughout its entire evolution and does not contribute significantly to the abundance of any element. It thus seems very unlikely that the simplest model of the NDW can produce the r-process. At most, it contributes to the production of the N = 50 closed shell elements and some light p-nuclei. In doing so, it may have left a distinctive signature on the abundances in metal-poor stars, but the results are sensitive to both uncertain models for the explosion and the masses of the neutron stars involved.