Low temperature scanning tunneling spectroscopy reveals the local density of states of the organic superconductor $kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Br, that was cut in-situ in ultra-high vacuum perpendicular to the superconducting BEDT-TTF layers. The spectra confirm that superconductivity is confined to the conducting BEDT-TTF layers, while the Cu[N(CN)$_2$]Br anion layers are insulating. The density of states comprises a twofold superconducting gap, which is attributed to the two separated bands crossing the Fermi surface.