Recent ideas based on the properties of assemblies of frictionless particles in mechanical equilibrium provide a perspective of amorphous systems different from that offered by the traditional approach originating in liquid theory. The relation, if any, between these two points of view, and the relevance of the former to the glass phase, has been difficult to ascertain. In this Letter, we introduce a model for which both theories apply strictly: it exhibits on the one hand an ideal glass transition and on the other "jamming" features (fragility, soft modes) virtually identical to that of real systems. This allows us to disentangle the two physical phenomena.