On the basis of its low luminosity, its chemical composition, and the absence of a large-scale outflow, the dense core Cha-MMS1 located in the Chamaeleon I molecular cloud was proposed as a first hydrostatic core (FHSC) candidate a decade ago. Our goal is to test this hypothesis by searching for a slow, compact outflow driven by Cha-MMS1 that would match the predictions of MHD simulations for this short phase of star formation. We use the Atacama Large Millimeter/submillimeter Array (ALMA) to map Cha-MMS1 at high angular resolution in CO 3-2 and ^13^CO 3-2 as well as in continuum emission. We report the detection of a bipolar outflow emanating from the central core, along a (projected) direction roughly parallel to the filament in which Cha-MMS1 is embedded and perpendicular to the large-scale magnetic field. The morphology of the outflow indicates that its axis lies close to the plane of the sky. We measure velocities corrected for inclination of more than 90km/s which is clearly incompatible with the expected properties of a FHSC outflow. Several properties of the outflow are determined and compared to previous studies of Class 0 and Class I protostars. The outflow of Cha-MMS1 has a much smaller momentum force than the outflows of other Class0 protostars. In addition, we find a dynamical age of 200-3000yr indicating that Cha-MMS1 might be one of the youngest ever observed Class 0 protostars. While the existence of the outflow suggests the presence of a disk, no disk is detected in continuum emission and we derive an upper limit of 55au to its radius. We conclude that Cha-MMS1 has already gone through the FHSC phase and is a young Class 0 protostar, but it has not brought its outflow to full power yet.