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 ± 0.016 M ⊙ and a radius of R ⋆ = 0.382 ± 0.012 R ⊙. 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).