The interplay of the massless Dirac fermions in graphene and the Cooper pair states in a superconductor has the potential to give rise to exotic physical phenomena and useful device applications. But to date, the junctions formed between graphene and superconductors on conventional substrates have been highly disordered. Charge scattering and potential fluctuations caused by such disorder are believed to have prevented the emergence or observation of new physics. Here we propose to address this problem by forming suspended graphene-superconductor junctions. We demonstrate the fabrication of high-quality suspended monolayer graphene-NbN Josephson junctions with device mobility in excess of 150,000 cm(2) per Vs, minimum carrier density below 10(10) cm(-2), and the flow of a supercurrent at critical temperatures greater than 2 K. The characteristics of our Josephson junctions are consistent with ballistic transport, with a linear dependence on the Fermi energy that reflects of linear dispersion of massless Dirac fermions.