Graphene—a recently discovered form of graphite only one atomic layer thick1—constitutes a new model system in condensed matter physics, because it is the first material in which charge carriers behave as massless chiral relativistic particles. The anomalous quantization of the Hall conductance2, 3, which is now understood theoretically4, 5, is one of the experimental signatures of the peculiar transport properties of relativistic electrons in graphene. Other unusual phenomena, like the finite conductivity of order 4e2/h (where e is the electron charge and h is Planck's constant) at the charge neutrality (or Dirac) point2, have come as a surprise and remain to be explained5, 6, 7, 8, 9, 10, 11, 12, 13. Here we experimentally study the Josephson effect14 in mesoscopic junctions consisting of a graphene layer contacted by two closely spaced superconducting electrodes15. The charge density in the graphene layer can be controlled by means of a gate electrode. We observe a supercurrent that, depending on the gate voltage, is carried by either electrons in the conduction band or by holes in the valence band. More importantly, we find that not only the normal state conductance of graphene is finite, but also a finite supercurrent can flow at zero charge density. Our observations shed light on the special role of time reversal symmetry in graphene, and demonstrate phase coherent electronic transport at the Dirac point.