We have calculated synchrotron spectra of relativistic blast waves, and find predicted characteristic frequencies that are more than an order of magnitude different from previous calculations. For the case of an adiabatically expanding blast wave, which is applicable to observed gamma-ray burst (GRB) afterglows at late times, we give expressions to infer the physical properties of the afterglow from the measured spectral features. We show that enough data exist for GRB 970508 to compute unambiguously the ambient density, n=0.04, and the blast wave energy per unit solid angle, E=4.e52 erg/(4pi sterad) We also compute the energy density in electrons and magnetic field. We find that they are 13% and 7%, respectively, of the nucleon energy density and thus confirm for the first time that both are close to but below equipartition. For GRB 971214, we show that the break found in its spectrum by Ramaprakash et al. (1998) is unlikely to be the synchrotron peak frequency, but could be the cooling break. We argue that the peak frequency was spotted in early IR measurements (Gorosabel et al. 1998). Using the three available constraints, we show that this afterglow has a lower energy, but the fractions transferred to electrons and magnetic field are consistent with those found in GRB 970508. The inferred gamma-ray to afterglow luminosity ratio is rather different between these two bursts, which we attribute to either intrinsic beaming of the gamma rays, or to a longer initial radiative phase in GRB 971214.