The European Spallation Source, currently under construction in Lund, Sweden, is a multidisciplinary international laboratory. Once completed to full specifications, it will operate the world's most powerful pulsed neutron source. Supported by a 3 million Euro Research and Innovation Action within the EU Horizon 2020 program, a design study (HighNESS) has been completed to develop a second neutron source located below the spallation target. Compared to the first source, designed for high cold and thermal brightness, the new source has been optimized to deliver higher intensity, and a shift to longer wavelengths in the spectral regions of cold (CN, 2-20 angstrom), very cold (VCN, 10-120 angstrom), and ultracold (UCN, >500 angstrom) neutrons. The second source comprises a large liquid deuterium moderator designed to produce CN and support secondary VCN and UCN sources. Various options have been explored in the proposed designs, aiming for world-leading performance in neutronics. These designs will enable the development of several new instrument concepts and facilitate the implementation of a high-sensitivity neutron-antineutron oscillation experiment (NNBAR). This document serves as the Conceptual Design Report for the HighNESS project, representing its final deliverable.