Diamagnetic exciton spectra of InSb crystals derived from transmission measurements carried out on thin unstressed samples a t 1.8 K and in magnetic fields n p t o 80 kOe are used for a direct determination of parameters of the c-and v-bands. Detailed identification and precise binding energy calculations permit one to derive the cyclotron frequencies of the electron and the holes, and the spin splittings. The m?(eC) and g:(Ec) relationships thus obtained exhibit good agreement with Knne's model up to energies close to E~ Y cg. Light holes also reveal a strong nonparabolicity characterized in the linear-in-k4 approximation by the coefficient 'plh = 0.47 meV-l. An analysis of the splitting of two light-hole long-wavelength transitions including the diamagnetic exciton binding energies and the data on the c-band obtained permits one to calculate the v-band parameter set.