First-order, or discontinuous, synchronization transition, i.e. an abrupt and irreversible phase transition with hysteresis to the synchronized state of coupled oscillators, has attracted much attention along the past years. We here report the analytical solution of a generalized Kuramoto model, and derive a series of exact results for the first-order synchronization transition, including i) the exact, generic, solutions for the critical coupling strengths for both the forward and backward transitions, ii) the closed form of the forward transition point and the linear stability analysis for the incoherent state (for a Lorentzian frequency distribution), and iii) the closed forms for both the stable and unstable coherent states (and their stabilities) for the backward transition. Our results, together with elucidating the first-order nature of the transition, provide insights on the mechanisms at the basis of such a synchronization phenomenon.