We report on the unusual nature of the ν=0 state in the integer quantum Hall effect (QHE) in graphene and show that electron transport in this regime is dominated by counterpropagating edge states. Such states, intrinsic to massless Dirac quasiparticles, manifest themselves in a large longitudinal resistivity ρxx h/e2, in striking contrast to ρxx behavior in the standard QHE. The ν=0 state in graphene is also predicted to exhibit pronounced fluctuations in ρxy and ρxx and a smeared zero Hall plateau in σxy, in agreement with experiment. The existence of gapless edge states puts stringent constraints on possible theoretical models of the ν=0 state. © 2007 The American Physical Society.