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_⋆ = 0.379 ± 0.016M_⊙ and a radius of R_⋆ = 0.382 ± 0.012R_⊙. The ultra-short-period planet LTT 3780 b (P_b = 0.77 d) with a radius of 1.35_−0.06^+0.06 R_⊕, a mass of 2.34^−0.23_+0.24 M_⊕, and a bulk density of 5.24_−0.81^+0.94 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_⊕, mass of 6.29_−0.61^+0.63 M_⊕, and mean density of 2.45_−0.37^+0.44 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 anexcellent 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 (JWST).