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Full list of authors: Nowak, G.; Luque, R.; Parviainen, H.; Pallé, E.; Molaverdikhani, K.; Béjar, V. J. S.; Lillo-Box, J.; Rodríguez-López, C.; Caballero, J. A.; Zechmeister, M.; Passegger, V. M.; Cifuentes, C.; Schweitzer, A.; Narita, N.; Cale, B.; Espinoza, N.; Murgas, F.; Hidalgo, D.; Zapatero Osorio, M. R.; Pozuelos, F. J.; Aceituno, F. J.; Amado, P. J.; Barkaoui, K.; Barrado, D.; Bauer, F. F.; Benkhaldoun, Z.; Caldwell, D. A.; Casasayas Barris, N.; Chaturvedi, P.; Chen, G.; Collins, K. A.; Collins, K. I.; Cortés-Contreras, M.; Crossfield, I. J. M.; de León, J. P.; Díez Alonso, E.; Dreizler, S.; El Mufti, M.; Esparza-Borges, E.; Essack, Z.; Fukui, A.; Gaidos, E.; Gillon, M.; Gonzales, E. J.; Guerra, P.; Hatzes, A.; Henning, Th.; Herrero, E.; Hesse, K.; Hirano, T.; Howell, S. B.; Jeffers, S. V.; Jehin, E.; Jenkins, J. M.; Kaminski, A.; Kemmer, J.; Kielkopf, J. F.; Kossakowski, D.; Kotani, T.; Kürster, M.; Lafarga, M.; Latham, D. W.; Law, N.; Lissauer, J. J.; Lodieu, N.; Madrigal-Aguado, A.; Mann, A. W.; Massey, B.; Matson, R. A.; Matthews, E.; Montañés-Rodríguez, P.; Montes, D.; Morales, J. C.; Mori, M.; Nagel, E.; Oshagh, M.; Pedraz, S.; Plavchan, P.; Pollacco, D.; Quirrenbach, A.; Reffert, S.; Reiners, A.; Ribas, I.; Ricker, G. R.; Rose, M. E.; Schlecker, M.; Schlieder, J. E.; Seager, S.; Stangret, M.; Stock, S.; Tamura, M.; Tanner, A.; Teske, J.; Trifonov, T.; Twicken, J. D.; Vanderspek, R.; Watanabe, D.; Wittrock, J.; Ziegler, C.; Zohrabi, F. ; 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 Teff = 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 (Pb = 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). © 2020 ESO. ; CARMENES is an instrument for the Centro Astronomico Hispano-Aleman de Calar Alto (CAHA, Almeria, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Cientificas (CSIC), the European Union through FEDER/ERF FICTS-2011-02 funds, and the members of the CARMENES Consortium (Max-Planck-Institut fur Astronomie, Instituto de Astrofisica de Andalucia, Landessternwarte Konigstuhl, Institut de Ciencies de l'Espai, Institut fur Astrophysik Gottingen, Universidad Complutense de Madrid, Thuringer Landessternwarte Tautenburg, Instituto de Astrofisica de Canarias, Hamburger Sternwarte, Centro de Astrobiologia and Centro Astronomico Hispano-Aleman), with additional contributions by the Spanish Ministry of Economy, the German Science Foundation through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 "Blue Planets around Red Stars", the Klaus Tschira Stiftung, the states of Baden-Wurttemberg and Niedersachsen, and by the Junta de Andalucia. This paper includes data collected by the TESS mission. Funding for the TESS mission is provided by the NASA Explorer Program. We acknowledge the use of public TOI Release data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This article is partly based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos Sanchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide. This work makes use of observations from the LCOGT network. This work makes use of observations acquired with the T150 telescope at Sierra Nevada Observatory, operated by the Instituto de Astrofisica de Andalucia (IAACSIC). Some of the Observations in the paper made use of the High-Resolution Imaging instrument 'Alopeke at Gemini-North. `Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. IRD is operated by the Astrobiology Center of the National Institutes of Natural Sciences. The research leading to these results has received funding from the ARC grant for Concerted Research Actions, financed by the WalloniaBrussels Federation. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Fundation (SNF). TRAPPIST-North is a project funded by the University of Liege (Belgium), in collaboration with Cadi Ayyad University of Marrakech (Morocco) M.G. and E.J. are F.R.S.-FNRS Senior Research Associate. The authors acknowledge funding from the Spanish Ministry of Economics and Competitiveness through projects PGC2018-098153-B-C31 and AYA2015-69350-C3-2-P. This work is partly supported by JSPS KAKENHI Grant Numbers JP18H01265 and JP18H05439, and JST PRESTO Grant Number JPMJPR1775. V.M.P. acknowledges support from NASA Grant NNX17AG24G. T.H. acknowledges support from the European Research Council under the Horizon 2020 Framework Program via the ERC Advanced Grant Origins 83 24 28. This research has been partially funded by Project No. MDM-2017-0737 Unidad de Excelencia "Maria de Maeztu" -Centro de Astrobiologia (INTA-CSIC). This research acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofisica de Andalucia (SEV-2017-0709) and project AYA2016-794425.