  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.
