We propose an EASY (Electroluminescent ApparatuS of high Yield) and SOFT (Separated Optimized FuncTion) time-projection chamber for the NEXT experiment, that will search for neutrinoless double beta decay (bb0nu) in Xe-136. Our experiment must be competitive with the new generation of bb0nu searches already in operation or in construction. This requires a detector with very good energy resolution (<1%), very low background con- tamination (1E-4 counts/(keV bullet kg bullet y)) and large target mass. In addition, it needs to be operational as soon as possible. The design described here optimizes energy resolution thanks to the use of proportional electroluminescent amplification (EL); it is compact, as the Xe gas is under high pressure; and it allows the measurement of the topological signature of the event to further reduce the background contamination. The SOFT design uses different sensors for tracking and calorimetry. We propose the use of SiPMs (MPPCs) coated with a suitable wavelength shifter for the tracking, and the use of radiopure photomultipliers for the measurement of the energy and the primary scintillation needed to estimate the t0. This design provides the best possible energy resolution compared with other NEXT designs based on avalanche gain devices. The baseline design is an Asymmetric Neutrino Gas EL apparatus (ANGEL), which was already outlined in the NEXT LOI. ANGEL is conceived to be easy to fabricate. It requires very little R&D and most of the proposed solutions have already been tested in the NEXT-1 prototypes. Therefore, the detector can be ready by 2013. In this Conceptual Design Report (CDR) we discuss first the physics case, present a full design of the detector, describe the NEXT-1 EL prototypes and their initial results, and outline a project to build a detector with 100 kg of enriched xenon to be installed in the Canfranc Underground Laboratory in 2013.