We have obtained new images and high-resolution (R~22,400) near-infrared (1.2400-1.2575 μm) spectra of each component of the brown dwarf binary GJ 569 Bab using the adaptive optics facility of the Keck II telescope and the NIRSPEC spectrometer. These data have allowed us to improve the determination of the astrometric orbit and to measure radial velocities of the components. We have used the astrometric and spectroscopic measurements to derive the dynamical mass of each brown dwarf and the systemic velocity of the pair by means of a χ2 fitting technique. From various considerations, the mass of each component is likely in the range 0.034-0.070 Msolar (GJ 569 Bb) and 0.055-0.087 Msolar (GJ 569 Ba). This implies that the mass ratio q of the binary is greater than 0.4, the most likely value being q=0.75-0.85. Adopting 0.072 Msolar as the most conservative location of the substellar limit for solar metallicity, our analysis confirms GJ 569 Bb as the first genuine brown dwarf known without any theoretical assumptions. We have compared the dynamical masses of GJ 569 Ba and Bb, and their effective temperatures and luminosities, to the predictions of state-of-the-art theoretical evolutionary isochrones, finding that models exhibit good performance in the regime of high substellar masses if the binary is about a few hundred million years old. However, the surface gravities of GJ 569 Ba (M8.5 V) and Bb (M9 V) derived from our spectral analysis (the observed data have been compared to the latest synthetic spectra) appear to be smaller than the values provided by the evolutionary models.