A general method is presented for computing the abundances produced by objects which explode from states of very high temperature. The nuclear evolution of non-degenerate matter expanding from temperatures of at least 1010 K is studied in detail for various values of the expansion rate and of the proton- neutron abundance difference and baryon density at 1010 K. These three parameters are the only factors which can influence element production under the conditions considered. Definite abundance predictions are made for each of the lighter nuclei (A < 27) plus the sum of all others (A > 28), but only individual volume elements within the object are considered. It is found that a range of models characterized by initial proton-neutron equality (mass-fraction differences <1 ) and expansion rates faster than 10 times the "gravitational" rate V(24ir&p) can produce abundances which are in rough agreement with those observed in the most metal-weak stars in our Galaxy, if they contain appreciable helium. These same models produce abundances of i, 9Be, 10B, and 11B which are much closer to solar-system values, however, making the observation of these elements in old stars a possible test for the occurrence of violent events early in the history of the Galaxy. Detailed comparisons with observation, however, must await more measurements of both the cross- sections of many critical reactions and the element abundances, especially in the oldest stars and quasi- stellar objects. Apart from the question of whether the particular models considered are realized in nature, the general process appears to be promising as a method of synthesis, since the ratios among many of the elements and isotopes produced are close to those observed in the solar system. The major characteristic of this process, which makes possible the simultaneous synthesis of all stable nuclei (with the exception of 6Li) in the mass range considered, is the high abundance of neutrons, protons, and a-particles throughout. The very fast expansion rates and initial proton-neutron equality are necessary to guarantee continued proton-neutron equality, keeping the synthesis routes near the p-stability line.