We present the discovery and characterisation of two transiting planets observed by the Transiting Exoplanet Survey Satellite (TESS) orbiting the nearby (d ~ 22 pc), bright (J ~ 9 mag) M3.5 dwarf LTT 3780 (TOI-732). We confirm both planets and their association with LTT 3780 via ground-based photometry and determine their masses using precise radial velocities measured with the CARMENES spectrograph. Precise stellar parameters determined from CARMENES high resolution spectra confirm that LTT 3780 is a mid-M dwarf with an effective temperature of T_eff = 3360 +- 51 K, a surface gravity of log(g) = 4.81 +/- 0.04 (cgs), and an iron abundance of [Fe/H] = 0.09 +/- 0.16 dex, with an inferred mass of M_star = 0.379 +/- 0.016 M_sun and a radius of R_star = 0.382 +/- 0.012 R_sun. The ultra-short-period planet LTT 3780 b (P_b = 0.77 d) with a radius of 1.35^{+0.06}_{-0.06} R_earth, a mass of 2.34^{+0.24}_{-0.23} M_earth, and a bulk density of 5.24^{+0.94}_{-0.81} g cm^{-3} joins the population of Earth-size planets with rocky, terrestrial composition. The outer planet, LTT 3780 c, with an orbital period of 12.25 d, radius of 2.42^{+0.10}_{-0.10} R_earth, mass of 6.29^{+0.63}_{-0.61} M_earth, and mean density of 2.45^{+0.44}_{-0.37} g cm^{-3} belongs to the population of dense sub-Neptunes. With the two planets located on opposite sides of the radius gap, this planetary system is an excellent target for testing planetary formation, evolution and atmospheric models. In particular, LTT 3780 c is an ideal object for atmospheric studies with the James Webb Space Telescope.
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