We present a dynamical study of the Galactic black hole binary system Nova Muscae 1991 (GS/GRS 1124–683). We utilize 72 high-resolution Magellan Echellette spectra and 72 strictly simultaneous V-band photometric observations; the simultaneity is a unique and crucial feature of this dynamical study. The data were taken on two consecutive nights and cover the full 10.4 hr orbital cycle. The radial velocities of the secondary star are determined by cross-correlating the object spectra with the best-match template spectrum obtained using the same instrument configuration. Based on our independent analysis of five orders of the echellette spectrum, the semi-amplitude of the radial velocity of the secondary is measured to be {{K}2}=406.8+/- 2.7 km s-1, which is consistent with previous work, while the uncertainty is reduced by a factor of 3. The corresponding mass function is f(M)=3.02+/- 0.06 {{M}? }. We have also obtained an accurate measurement of the rotational broadening of the stellar absorption lines (vsin i=85.0+/- 2.6 km s-1), and hence the mass ratio of the system q=0.079+/- 0.007. Finally, we have measured the spectrum of the non-stellar component of emission that veils the spectrum of the secondary. In a future paper, we will use our veiling-corrected spectrum of the secondary and accurate values of K2 and q to model multi-color light curves and determine the systemic inclination and the mass of the black hole. This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.
