We solve the Dirac-Bogoliubov-de Gennes equation in an impurity-free superconductor-normal-metalsuperconductor junction, to determine the maximal supercurrent I c that can flow through an undoped strip of graphene with heavily doped superconducting electrodes. The result I c Ӎ͑W / L͒e⌬ 0 / ប is determined by the superconducting gap ⌬ 0 and by the aspect ratio of the junction ͑length L small relative to the width W and to the superconducting coherence length͒. Moving away from the Dirac point of zero doping, we recover the usual ballistic result I c Ӎ͑W / F ͒e⌬ 0 / ប, in which the Fermi wavelength F takes over from L. The product I c R N Ӎ ⌬ 0 / e of the critical current and normal-state resistance retains its universal value ͑up to a numerical prefactor͒ on approaching the Dirac point. DOI: 10.1103/PhysRevB.74.041401 PACS number͑s͒: 73.23.Ad, 74.45.ϩc, 74.50.ϩr, 74.78.Na While the Josephson effect was originally discovered in a tunnel junction, 1 any weak link between two superconductors can support a dissipationless current in equilibrium.2 The current I͑͒ varies periodically with the phase difference of the pair potential in the two superconductors, reaching a maximum I c ͑the critical current͒ which is characteristic of the strength of the link. A measure of the coupling strength is the resistance R N of the junction when the superconductors are in the normal state. The product I c R N increases as the separation L of the two superconductors becomes smaller and smaller, until it saturates at a value of order ⌬ 0 / e, determined only by the excitation gap ⌬ 0 in the superconductors-but independent of the coupling strength. This phenomenology has been well established in a variety of superconductor-normal-metal-superconductor ͑SNS͒ junctions 3 and forms the basis of operation of the Josephson field-effect transistor. 4,5 A new class of weak links has now become available for research, 6 in which the superconductors are coupled by a monatomic layer of carbon ͑graphene͒. The low-lying excitations in this material are described by a relativistic wave equation, the Dirac equation. They are massless, having a velocity v that is independent of energy, and gapless, occupying conduction and valence bands that touch at discrete points ͑Dirac points͒ in reciprocal space.7 Graphene thus provides a unique opportunity to explore the physics of the "relativistic Josephson effect" ͑which had remained unexplored in earlier work 8 on relativistic effects in hightemperature and heavy-fermion superconductors͒. We address this problem here in the framework of the DiracBogoliubov-de Gennes ͑DBdG͒ equation of Ref. 9.The basic question that we seek to answer is what happens to the critical current as we approach the Dirac point of zero carrier concentration. Earlier theories [11][12][13] have found that undoped graphene has a quantum-limited conductivity of order e 2 / h, in the absence of any impurities or lattice defects. We find that the critical current is given, up to numerical coefficients of order unity, byin the short-jun...