We report on a combined theoretical and experimental investigation of the superconducting state in the quasi-two-dimensional organic superconductor κ-(ET)_{2}Cu[N(CN)_{2}]Br. Applying spin-fluctuation theory to a low-energy, material-specific Hamiltonian derived from ab initio density functional theory we calculate the quasiparticle density of states in the superconducting state. We find a distinct three-peak structure that results from a strongly anisotropic mixed-symmetry superconducting gap with eight nodes and twofold rotational symmetry. This theoretical prediction is supported by low-temperature scanning tunneling spectroscopy on in situ cleaved single crystals of κ-(ET)_{2}Cu[N(CN)_{2}]Br with the tunneling direction parallel to the layered structure.