The authors present high signal-to-noise ratio spectrophotometry obtained in the optical and near-infrared with the double spectrograph ISIS on the WHT, of the M33 giant H II region NGC 604. The authors' main aim was to deduce global properties of a nearby giant H II region. The near-infrared spectral range, central to the authors' study, comprises the gravity-sensitive Ca II triplet (CaT) stellar absorption features. The data were obtained using two different techniques: (1) scanning with a 1.75-arcmin long slit that provided integrated spectra of the central part of the nebula, and (2) long-slit spectra of the brightest continuum knots. A single red supergiant (RSG) star was detected, judging from the measured strength of the stellar CaT absorption lines, visible in one region and only after carefully subtracting the hydrogen Paschen emission lines that dominate the near-infrared spectra. The feature is not observed in the integrated spectrum. The observed Paschen discontinuity in emission allows the authors to determine an electron temperature, which is similar to the one obtained from the ratio of forbidden line ([O III]) intensities, suggesting that temperature fluctuations are not present in the nebula. Wolf-Rayet (WR) features have been found in several positions; the observed He II λ4686-Å line intensity is found to be larger than in galactic WR stars, by a factor of ≡2. Exceptionally broad components of permitted lines of hydrogen and helium (FWHM ≡2500 km s-1) are observed in one of the brightest stars in NGC 604. This object shows a large overabundance of He and strong spectral variability on time-scales of ≡10 yr. The authors classify it as an LBV-WR transition object. They identify the core of the cluster ionizing the nebula, which is probably younger than 5 Myr. Implications of these results for the evolutionary state of NGC 604 are discussed. Also discussed are the difficulties involved in the techniques for using RSGs as tracers of starburst activity in galactic nuclei.