The Sachdev-Ye-Kitaev (SYK) model is a disordered quantum mean-field model studied in condensed matter physics and the holographic theory of black holes. Its structural properties can be derived heuristically using a combination of the replica method and path integration techniques. Analyzing it mathematically rigorously, however, turned out to be notoriously difficult, even for basic questions such as computing the annealed free energy. In this paper we rigorously compute the free energy limit (annealed and quenched) for this model at high enough but constant temperature. Our results are in numerical agreement with the results derived by physics methods. Remarkably, though, our method of proof is novel and is different from the physics approach. It is based on (a) the theory of the component structure of sparse random graphs and (b) a variant of the cavity method, used widely in prior rigorous and heuristic treatments of classical spin glasses.