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Gaia Data Release 3 (DR3) contains the first release of magnitudes estimated from the integration of Radial Velocity Spectrometer (RVS) spectra for a sample of about 32.2 million stars brighter than G_RVS~14 mag (or G~15 mag). In this paper, we describe the data used and the approach adopted to derive and validate the G_RVS magnitudes published in DR3. We also provide estimates of the G_RVS passband and associated G_RVS zero-point. We derived G_RVS photometry from the integration of RVS spectra over the wavelength range from 846 to 870 nm. We processed these spectra following a procedure similar to that used for DR2, but incorporating several improvements that allow a better estimation of G_RVS. These improvements pertain to the stray-light background estimation, the line spread function calibration, and the detection of spectra contaminated by nearby relatively bright sources. We calibrated the G_RVS zero-point every 30 hours based on the reference magnitudes of constant stars from the Hipparcos catalogue, and used them to transform the integrated flux of the cleaned and calibrated spectra into epoch magnitudes. The G_RVS magnitude of a star published in DR3 is the median of the epoch magnitudes for that star. We estimated the G_RVS passband by comparing the RVS spectra of 108 bright stars with their flux-calibrated spectra from external spectrophotometric libraries. The G_RVS magnitude provides information that is complementary to that obtained from the G, G_BP, and G_RP magnitudes, which is useful for constraining stellar metallicity and interstellar extinction. The median precision of G_RVS measurements ranges from about 0.006 mag for the brighter stars (i.e. with 3.5 < G_RVS < 6.5 mag) to 0.125 mag at the faint end. The derived G_RVS passband shows that the effective transmittance of the RVS is approximately 1.23 times better than the pre-launch estimate.