  We report the discovery of a planetary system orbiting TOI-763 (aka CD-39
7945), a $V=10.2$, high proper motion G-type dwarf star that was
photometrically monitored by the TESS space mission in Sector 10. We obtain and
model the stellar spectrum and find an object slightly smaller than the Sun,
and somewhat older, but with a similar metallicity. Two planet candidates were
found in the light curve to be transiting the star. Combining TESS transit
photometry with HARPS high-precision radial velocity follow-up measurements
confirm the planetary nature of these transit signals. We determine masses,
radii, and bulk densities of these two planets. A third planet candidate was
discovered serendipitously in the radial velocity data. The inner transiting
planet,TOI-763 b, has an orbital period of $P_\mathrm{b}$ = 5.6~days, a mass of
$M_\mathrm{b}$ = $9.8\pm0.8$ $M_\oplus$, and a radius of $R_\mathrm{b}$ =
$2.37\pm0.10$ $R_\oplus$. The second transiting planet,TOI-763 c, has an
orbital period of $P_\mathrm{c}$ = 12.3~days, a mass of $M_\mathrm{c}$ =
$9.3\pm1.0$ $M_\oplus$, and a radius of $R_\mathrm{c}$ = $2.87\pm0.11$
$R_\oplus$. We find the outermost planet candidate to orbit the star with a
period of $\sim$48~days. If confirmed as a planet it would have a minimum mass
of $M_\mathrm{d}$ = $9.5\pm1.6$ $M_\oplus$. We investigated the TESS light
curve in order to search for a mono transit by planet~d without success. We
discuss the importance and implications of this planetary system in terms of
the geometrical arrangements of planets orbiting G-type stars.
