Context. HD 93129A was classified as the earliest O-type star in the Galaxy (O2 If*) and is considered as the prototype of its spectral class. However, interferometry shows that this object is a binary system, while recent observations even suggest a triple configuration. None of the previous spectral analyses of this object accounted for its multiplicity. With new high-resolution UV and optical spectra, we have the possibility to reanalyze this key object, taking its binary nature into account for the first time. Aims: We aim to derive the fundamental parameters and the evolutionary status of HD 93129A, identifying the contributions of both components to the composite spectrum Methods: We analyzed UV and optical observations acquired with the Hubble Space Telescope and ESO’s Very Large Telescope. A multiwavelength analysis of the system was performed using the latest version of the Potsdam Wolf-Rayet model atmosphere code. Results: Despite the similar spectral types of the two components, we are able to find signatures from each of the components in the combined spectrum, which allows us to estimate the parameters of both stars. We derive log(L/L⊙) = 6.15, Teff = 52 kK, and log Ṁ = -4.7 [M⊙ yr-1] for the primary Aa, and log(L/L⊙) = 5.58, Teff = 45 kK, and log Ṁ = -5.8 [M⊙yr-1] for the secondary Ab. Conclusions: Even when accounting for the binary nature, the primary of HD 93129A is found to be one of the hottest and most luminous O stars in our Galaxy. Based on the theoretical decomposition of the spectra, we assign spectral types O2 If* and O3 III(f*) to components Aa and Ab, respectively. While we achieve a good fit for a wide spectral range, specific spectral features are not fully reproduced. The data are not sufficient to identify contributions from a hypothetical third component in the system. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained (from the Data Archive) at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program GO-13346.Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 095.D-0234(A).