We present spectroscopic observations of the quiescent black hole binary A0620-00 with the the 6.5-m Magellan Clay telescope at Las Campanas Observatory . We measure absorption-line radial velocities of the secondary and make the most precise determination to date ( K _ { 2 } = 435.4 \pm 0.5 km s ^ { -1 } ) . By fitting the rotational broadening of the secondary , we refine the mass ratio to q = 0.060 \pm 0.004 ; these results , combined with the orbital period , imply a minimum mass for the compact object of 3.10 \pm 0.04 M _ { \sun } . Although quiescence implies little accretion activity , we find that the disc contributes 56 \pm 7 per cent of the light in B and V , and is subject to significant flickering . Doppler maps of the Balmer lines reveal bright emission from the gas stream-disc impact point and unusual crescent-shaped features . We also find that the disc centre of symmetry does not coincide with the predicted black hole velocity . By comparison with SPH simulations , we identify this source with an eccentric disc . With high S/N , we pursue modulation tomography of H \alpha and find that the aforementioned bright regions are strongly modulated at the orbital period . We interpret this modulation in the context of disc precession , and discuss cases for the accretion disc evolution .