We report a combined theoretical and experimental investigation of the superconducting state in the quasi-two-dimensional organic superconductor $kappa$-(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 $kappa$-(ET)$_2$Cu[N(CN)$_2$]Br with the tunneling direction parallel to the layered structure.
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