  The Gaia mission has provided us full astrometric solutions for over 1.5B
sources. However, only the brightest 34M of those have radial velocity
measurements. As a proof of concept, this paper aims to close that gap, by
obtaining radial velocity estimates from the low-resolution BP/RP spectra that
Gaia now provides. These spectra are currently published for about 220M
sources, with this number increasing to the full $\sim 2$B Gaia sources with
Gaia Data Release 4. To obtain the radial velocity measurements, we fit Gaia
BP/RP spectra with models based on a grid of synthetic spectra, with which we
obtain the posterior probability on the radial velocity for each object. Our
measured velocities show systematic biases that depend mainly on colours and
magnitudes of stars. We correct for these effects by using external catalogues
of radial velocity measurements. We present in this work a catalogue of about
6.4M sources with our most reliable radial velocity measurements and
uncertainties $<300$ km s$^{-1}$ obtained from the BP/RP spectra. About 23% of
these have no previous radial velocity measurement in Gaia RVS. Furthermore, we
provide an extended catalogue containing all 125M sources for which we were
able to obtain radial velocity measurements. The latter catalogue, however,
also contains a fraction of measurements for which the reported radial
velocities and uncertainties are inaccurate. Although typical uncertainties in
the catalogue are significantly higher compared to those obtained with
precision spectroscopy instruments, the number of potential sources for which
this method can be applied is orders of magnitude higher than any previous
radial velocity catalogue. Further development of the analysis could therefore
prove extremely valuable in our understanding of Galactic dynamics.
