  Ultrasonic velocity and attenuation measurements on the quasi-two-dimensional
organic conductor $\kappa$-[BEDT-TTF]$_2$Cu[N(CN)$_2$]Br give the first
determination of the coexistence zone of the antiferromagnetic and
superconducting phases which extends deep in the metallic part of the
pressure-temperature phase diagram of these salts. This zone is identified from
a precise control of the ordering in ethylene end groups which acts as an
internal pressure. The two phases are found to compete whereas superconducting
fluctuations begin to contribute to the attenuation at 15 K, namely at the
onset of magnetic order that is well above the superconducting transition
temperature $T_c$ = 11.9 K. Finally, the temperature profile of sound
attenuation for both longitudinal and transverse phonon polarizations is found
to be inconsistent with a conventional \textit{s}-wave order parameter.
