We observed the brightest super-star cluster NGC 1705-1 in the nearby dwarf galaxy NGC 1705 with the Space Telescope Imaging Spectrograph in the echelle mode between 1200 and 3100 Å. The data allow a study of the young stellar population at hitherto unprecedented spectral resolution and signal-to-noise ratios. A comprehensive list of strong and weak stellar and interstellar absorption lines is given, together with the measured line parameters. Four distinct velocity systems are identified: stellar lines at the measured H I velocity, blueshifted interstellar lines from outflowing gas, Milky Way foreground absorption, and a high-velocity cloud. Comparison with stellar template spectra indicates an equivalent spectral type of B0 to B1, with mostly dwarf and giant stars contributing. When placed on a theoretical Hertzsprung-Russell diagram, these stars constrain the age of NGC 1705-1 to 12±13 Myr. Since this age is derived purely from spectroscopy, it is independent of reddening corrections. A comparison of the observed and theoretical mass-to-light ratio for the derived age was performed. We find no significant evidence for an anomalous initial mass function at the low-mass end, contrary to suggestions found in the literature. The stellar population of NGC 1705-1 is similar to that in other massive clusters, such as 30 Doradus or NGC 1569-A, after taking into account age differences and model uncertainties. We discuss the difficulty of relating observed and theoretical mass-to-light ratios because of the unknown gas mass fraction lost by the cluster and the uncertain mass-loss rates of asymptotic giant branch stars in population synthesis models.