Despite being only one-atom thick, defect-free graphene is considered to be completely impermeable to all gases and liquids1-10. This conclusion is based on theory3-8 and supported by experiments1,9,10 that could not detect gas permeation through micrometre-size membranes within a detection limit of 105 to 106 atoms per second. Here, using small monocrystalline containers tightly sealed with graphene, we show that defect-free graphene is impermeable with an accuracy of eight to nine orders of magnitude higher than in the previous experiments. We are capable of discerning (but did not observe) permeation of just a few helium atoms per hour, and this detection limit is also valid for all other gases tested (neon, nitrogen, oxygen, argon, krypton and xenon), except for hydrogen. Hydrogen shows noticeable permeation, even though its molecule is larger than helium and should experience a higher energy barrier. This puzzling observation is attributed to a two-stage process that involves dissociation of molecular hydrogen at catalytically active graphene ripples, followed by adsorbed atoms flipping to the other side of the graphene sheet with a relatively low activation energy of about 1.0 electronvolt, a value close to that previously reported for proton transport11,12. Our work provides a key reference for the impermeability of two-dimensional materials and is important from a fundamental perspective and for their potential applications.