  Dynamical histories of planetary systems, as well as atmospheric evolution of
highly irradiated planets, can be studied by characterizing the
ultra-short-period planet population, which the TESS mission is particularly
well suited to discover. Here, we report on the follow-up of a transit signal
detected in the TESS sector 19 photometric time series of the M3.0 V star
TOI-1685 (2MASS J04342248+4302148). We confirm the planetary nature of the
transit signal, which has a period of P_b=0.6691403+0.0000023-0.0000021 d,
using precise radial velocity measurements taken with the CARMENES
spectrograph. From the joint photometry and radial velocity analysis, we
estimate the following parameters for TOI-1685 b: a mass of M_b=3.78+/-0.63
M_Earth, a radius of R_b=1.70+/-0.07 R_Earth, which together result in a bulk
density of rho_b=4.21+0.95-0.82 g/cm3, and an equilibrium temperature of
Teq_b=1069+/-16 K. TOI-1685 b is the least dense ultra-short period planet
around an M dwarf known to date. TOI-1685 b is also one of the hottest
transiting Earth-size planets with accurate dynamical mass measurements, which
makes it a particularly attractive target for thermal emission spectroscopy.
Additionally, we report a further non-transiting planet candidate in the
system, TOI-1685[c], with an orbital period of P_[c]=9.02+0.10-0.12 d.
